Numerical Methods for Structured Matrices and Applications: The Georg Heinig Memorial Volume

Operator Theory: Advances and Applications Vol. 199 Founded in 1979 by Israel Gohberg Editors: Harry Dym (Rehovot, Isr...

Author: Dario Andrea Bini | Volker Mehrmann | Vadim Olshevsky | Eugene Tyrtsyhnikov | Marc van Barel

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Operator Theory: Advances and Applications Vol. 199 Founded in 1979 by Israel Gohberg

Editors: Harry Dym (Rehovot, Israel) Joseph A. Ball (Blacksburg, VA, USA) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Vienna, Austria) Christiane Tretter (Bern, Switzerland)

Associate Editors: Vadim Adamyan (Odessa, Ukraine) Albrecht Böttcher (Chemnitz, Germany) B. Malcolm Brown (Cardiff, UK) Raul Curto (Iowa City, IA, USA) Fritz Gesztesy (Columbia, MO, USA) Pavel Kurasov (Lund, Sweden) Leonid E. Lerer (Haifa, Israel) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Leiba Rodman (Williamsburg, VI, USA) Ilya M. Spitkovsky (Williamsburg, VI, USA)

Subseries Linear Operators and Linear Systems Subseries editors: Daniel Alpay (Beer Sheva, Israel) Birgit Jacob (Wuppertal, Germany) André C.M. Ran (Amsterdam, The Netherlands)

Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze (Potsdam, Germany) Michael Demuth (Clausthal, Germany) Jerome A. Goldstein (Memphis, TN, USA) Nobuyuki Tose (Yokohama, Japan)

Honorary and Advisory Editorial Board: Lewis A. Coburn (Buffalo, NY, USA) Ciprian Foias (College Station, TX, USA) J.William Helton (San Diego, CA, USA) Thomas Kailath (Stanford, CA, USA) Peter Lancaster (Calgary, AB, Canada) Peter D. Lax (New York, NY, USA) Donald Sarason (Berkeley, CA, USA) Bernd Silbermann (Chemnitz, Germany) Harold Widom (Santa Cruz, CA, USA)

Numerical Methods for Structured Matrices and Applications The Georg Heinig Memorial Volume

Dario Andrea Bini Volker Mehrmann Vadim Olshevsky Eugene E. Tyrtyshnikov Marc van Barel Editors

Birkhäuser

Editors: Dario Andrea Bini Dipartimento di Matematica Università di Pisa Largo Bruno Pontecorvo, 5 56127 Pisa Italy e-mail: [email protected]

Eugene E. Tyrtyshnikov Institute of Numerical Mathematics Russian Academy of Sciences Gubkina Street, 8 Moscow, 119991 Russia e-mail: [email protected]

Volker Mehrmann Institut für Mathematik Technische Universität Berlin Straße des 17. Juni 136 10623 Berlin Germany e-mail: [email protected]

Marc van Barel Katholieke Universiteit Leuven Department of Computer Science Celestijnenlaan 200A 3001 Leuven (Heverlee) Belgium e-mail: [email protected]

Vadim Olshevsky Department of Mathematics University of Connecticut 196 Auditorium Road, U-9 Storrs, CT 06269 USA e-mail: [email protected]

2010 Mathematics Subject Classiﬁcation: 15 Library of Congress Control Number: 2010920068

Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-0346-8995-6 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microﬁlms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2010 Birkhäuser / Springer Basel AG P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany ISBN 978-3-7643-8995-6

e-ISBN 978-3-7643-8996-3

987654321

www.birkhauser.ch

Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Part I: Georg Heinig A. B¨ ottcher, I. Gohberg and B. Silbermann Georg Heinig (1947–2005) In Memoriam . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

K. Rost Georg Heinig, November 24, 1947 – May 10, 2005 A Personal Memoir and Appreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

G. Heinig and K. Rost Introduction to Bezoutians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

T. Amdeberhan and G. Heinig On Matrices that are not Similar to a Toeplitz Matrix and a Family of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Part II: Research Contributions T. Bella, Y. Eidelman, I. Gohberg, V. Olshevsky, E. Tyrtyshnikov and P. Zhlobich A Traub-like Algorithm for Hessenberg-quasi-separableVandermonde Matrices of Arbitrary Order . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 D.A. Bini and P. Boito A Fast Algorithm for Approximate Polynomial GCD Based on Structured Matrix Computations . . . . . . . . . . . . . . . . . . . . . . . . . .

155

V. Bolotnikov On Inertia of Some Structured Hermitian Matrices . . . . . . . . . . . . . . . . . . . 175 A. B¨ ottcher and S. Grudsky Variable-coeﬃcient Toeplitz Matrices with Symbols beyond the Wiener Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 E. Bozzo and D. Fasino A Priori Estimates on the Structured Conditioning of Cauchy and Vandermonde Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

vi

Contents

V. Cort´es and J.M. Pe˜ na Factorizations of Totally Negative Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 221 S. Delvaux, L. Gemignani and M. Van Barel QR-factorization of Displacement Structured Matrices Using a Rank Structured Matrix Approach . . . . . . . . . . . . . . . . . . . . . . . . . .

229

S. Feldmann Bezoutians Applied to Least Squares Approximation of Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 B. Fritzsche, B. Kirstein and A. Lasarow On the Weyl Matrix Balls Corresponding to the Matricial Carath´eodory Problem in Both Nondegenerate and Degenerate Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

289

B. Fritzsche, B. Kirstein and L.A. Sakhnovich On Extremal Problems of Interpolation Theory with Unique Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 J. Jain, H. Li, C.-K. Koh and V. Balakrishnan O(n) Algorithms for Banded Plus Semiseparable Matrices . . . . . . . . . . . . 347 V.Y. Pan, B.J. Murphy and R.E. Rosholt Uniﬁed Nearly Optimal Algorithms for Structured Integer Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 C. Tablino Possio V-cycle Optimal Convergence for DCT-III Matrices . . . . . . . . . . . . . . . . . . 377 S.M. Rump and H. Sekigawa The Ratio Between the Toeplitz and the Unstructured Condition Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Y.V. Shlapak A New Algorithm for Finding Positive Eigenvectors for a Class of Nonlinear Operators Associated with M-matrices . . . . . . . . . . 421 E. Tyrtyshnikov Hankel Minors and Pade Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

Foreword Georg Heing, a charming, erudite man, and a ﬁrst rate mathematician died unexpectedly of a heart attack on May 10, 2005. Georg is survived by his wife Gerti, his daughter Susanne, and his son Peter. We have lost one the leading experts in the ﬁeld of structured matrices, a wonderful colleague, and a terriﬁc friend. Georg Heinig’s results, approaches, and his scientiﬁc taste inﬂuenced our community of researchers working on structured matrices. In fact, the community’s focus grew to reﬂect his interdisciplinary vision ranging from applications (e.g., in systems and control theory and signal processing) through fundamental mathematics (structured matrices, periodic Jacobi, Toeplitz, and Wiener-Hopf operators, classes of singular integral operators, resultants and Bezoutians for operator-valued polynomials and continual analogs thereof) to numerical analysis and fast algorithms. The broad spectrum of Georg Heinig’s interests are represented in this collection. Georg served as an Associate Editor of two top journals: Integral Equations and Operator Theory and Linear Algebra and Its Applications. This volume starts with two eulogies published earlier by IEOT and LAA. The ﬁrst one, published in IEOT is by Albrecht B¨ ottcher, Israel Gohberg (who was Georg’s advisor during his Ph.D. studies), and Bernd Silbermann. The second one, published in LAA is by Karla Rost who collaborated with Georg during last three decades until day of his death. They have produced together more than 30 papers and a monograph. We refer to these two eulogies for the details of Georg’s career, and here we would like to emphasize only one point, namely the inﬂuence of his work in the area of structured matrices. Matrices with structure (e.g., Toeplitz matrices) are encountered in a surprising variety of areas in sciences and mathematics. There were many approaches to study Toeplitz structure and its generalizations, one of them was known under the name “displacement structure method.” In their 1984 monograph “Algebraic Methods for Toeplitz-like Matrices and perators” G.Heinig and K.Rost demonstrated that this method (they called it the UV-reduction method) can be successfully used not only for Toeplitz structure and its derivatives, but also for many other patterns of structure, e.g., Hankel, Vandermonde, Cauchy matrices, Bezoutians and their generalizations. This breakthrough discovery facilitated a lot of interest in the community. Moreover, the new technique was immediately picked up and it was heavily used in the work of a number of research groups in Germany, USA, Israel, Leuven, Moscow, Hong Kong.

viii

Foreword

As Georg mentioned many times, about 20 years ago he was virtually alone delivering talks on structured matrices at such conferences as IWOTA and ILAS meetings. Nowadays special sessions and minisymposia on structured matrices are routinely included in programs of a number of conferences such as IWOTA, ILAS, SIAM annual meetings, SPIE, MTNS. Moreover, a number of conferences dedicated exclusively to structured matrices has been held (two AMS meetings in the USA, four conferences in Italy, three in Moscow, three in Hong Kong). Needless to say, Georg’s results, ideas, his energy, and service to the community facilitated this development and strongly inﬂuenced the research eﬀorts of structured matrices community. We are happy to include in this volume a joint paper of Georg Heinig and Karla Rost on Bezoutians. This is a subject Georg worked on since the very beginning of his career, and to which he made a number of signiﬁcant contributions. The paper blends an wonderful exposition of classical results with a survey recent development in the ﬁeld. It was a great honor and a privilege to edit this volume of papers dedicated in Georg’s memory. The Editors

Part I Georg Heinig

Operator Theory: Advances and Applications, Vol. 199, 3–6 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Georg Heinig (1947–2005) In Memoriam

On May 10, 2005, Georg Heinig died unexpectedly of a heart attack in his apartment in Kuwait. We have lost one of the top experts in the ﬁeld of structured matrices, an irreplaceable colleague, and a good friend. He was an active member of the editorial boards of the journal Integral Equations and Operator Theory and the book series Operator Theory: Advances and Applications since 1993. Our heartfelt condolences go out to his wife and his family. Originally published in Integr. equ. oper. theory 53 (2005), 297–300.

4

In Memoriam Georg Heinig

Georg Heinig was born on November 24, 1947 in the small town of Zschopau in the Ore Mountains (Erzgebirge) in East Germany. From 1954 to 1964 he attended the school in Zschopau and from 1964 to 1966 the elite class for mathematics at Chemnitz University of Technology. Such elite classes were established to provide especially gifted pupils with an extraordinary education in mathematics (but also in the natural sciences and in languages) under the guidance of experienced university teachers. The careers of many successful East German scientists started at elite classes. None of these classes has survived the German reuniﬁcation. He studied mathematics at Chemnitz University of Technology from 1966 to 1970 and graduated with the diploma degree in 1970. His diploma thesis was written under the supervision of Siegfried Pr¨ ossdorf and was devoted to certain properties of normally solvable operators in Banach spaces. After defending his diploma thesis with the best possible grade, Georg Heinig was given the opportunity of entering a PhD program abroad. He decided to continue his studies at Kishinev (now Chisinau) University under the supervision of the second of us. His wife Gerti accompanied him in Kishinev and also completed a dissertation during that period. Georg Heinig was a very talented and dedicated researcher. In Kishinev he embarked on research into the theory of Toeplitz, Wiener-Hopf, and singular integral operators with scalar and matrix-valued symbols, and it was during those wonderful years that he has fallen in love with all the exciting mathematics of structured matrices. His deep results in this area formed the basis of his excellent PhD thesis, which he defended in Spring of 1974. Many other mathematical insights gained by Georg during the years in Kishinev went into his habilitation thesis, which he completed in Chemnitz. The early paper Gohberg/Heinig, Inversion of ﬁnite Toeplitz matrices consisting of elements of a non-commutative algebra (Russian), Rev. Roumaine Math. Phys. Appl. 19, 623–663 (1974) became one of his most frequently cited works. Georg Heinig returned to Chemnitz in 1974. In the following ﬁve years the ﬁrst of us had the pleasure of attending his classes as a student, the third of us received an outstanding member of his research group, and the second of us was proud of Georg’s outstanding mathematical achievements. Georg Heinig integrated several young people into his research, Karla Rost being the most prominent ﬁgure of them. In 1979 he defended his habilitation thesis, which was on the spectral theory of operator bundles and the algebraic theory of ﬁnite Toeplitz matrices. His two children Peter and Susanne were born in 1974 and 1977. The scientiﬁc outcome of the research directed by Georg Heinig in the 1970s and early 1980s is summarized in his and Karla Rost’s book Algebraic Methods for Toeplitz-like Matrices and Operators, which was originally published by AkademieVerlag, Berlin in 1984 and was republished by Birkh¨ auser Verlag, Basel in the same year. This book has found a warm reception and perpetual interest by a large community for now about twenty years. Some of its basic ideas, such as the so-called U V reduction (which later received more popularity under the name displacement operation), have become important tools for workers in the ﬁeld of

In Memoriam Georg Heinig

5

structured matrices. Moreover, the scientiﬁc collaboration of Georg Heinig with Karla Rost lasted three decades until the day of Georg’s death. Their joint research resulted in more than 30 papers. The results and methods of these papers are an essential ingredient to the present-day mathematical high-technology one is encountering in connection with structured matrices. In 1982, Georg Heinig was a guest professor at Aleppo University in Syria, and from 1987 to 1989, he held a guest professorship at Addis Ababa University in Ethiopia. In the late 1980s he was appointed full professor at Leipzig University. After the political events in Germany at the turn to the 1990s the life for Georg changed dramatically. All people working at East German universities were formally dismissed and had to apply for a position anew. Those who had shown a certain extent of political proximity to the former socialist system had no chance of receiving a new position at a German university, neither in East Germany nor in the subsequently reuniﬁed Germany. The situation was extremely diﬃcult, and the eﬀorts of Georg’s friends to help him did not bring any positive results. Certainly Georg was very disappointed and despaired. Some time he planned to take over his father’s store for vegetables, but eventually he looked for a job at a foreign university. In 1993, Georg Heing went to Kuwait University, where he worked as a professor until his tragic death. The scientiﬁc conditions at Kuwait University were excellent and Georg has always thankfully acknowledged the recognition and friendship he received from his Kuwaiti colleagues. In 2003, he was awarded as the Scientist of the Year by the Amir of Kuwait. Despite all these successes, his and his wife’s dream was to endure the university job only until the age of 60 years and then simply to relish life together, including travelling around the world. His unexpected death at the age of 57 abruptly dispersed this dream. Georg Heinig’s scientiﬁc legacy is immense. In more than 100 publications he made outstanding contributions to a variety of ﬁelds, including – theory and fast algorithms for several classes of structured matrices, – periodic Jacobi, Toeplitz, and Wiener-Hopf operators, – classes of singular integral operators, – resultants and Bezoutians for operator-valued polynomials, – continual analogs of resultants and Bezoutians, – numerical methods for convolution equations, – applications in systems and control theory and signal processing. Discoveries by Georg and his co-workers, such as the structure of the kernel and of the pseudoinverse for certain classes of structured matrices, signiﬁcantly shaped the development of numerical algorithms. He also remarkably enriched various areas of operator theory, for example by deep results on the spectral theory of Jacobi matrices and of Toeplitz and Wiener-Hopf operators. He supervised 6 dissertations.

6

In Memoriam Georg Heinig

Georg Heing was a very pleasant person and an inspiring colleague. His sense of humor and his characteristic bright laughing will be missed by everyone who was lucky enough to meet him. His permanent endeavor for disclosing the absolute essence of a matter and his untiring aspiration for clearness and brevity were challenges for his co-workers on the one hand and have resulted in grateful appreciation by his students and the readers of his publications on the other. Another dream of Georg Heinig was a joint textbook with Karla Rost on structured matrices, ranging from the basics for beginners up to recent developments. About one year ago they started writing this book and three chapters are already more or less complete. It is unimaginable that he will never have this book in his hands some day. This tragedy bitterly reveals the gap that Georg has left and painfully reminds us of the projects and ideas that passed away with him. However, his work will endure and we will always remember this outstanding mathematician, excellent colleague, and wonderful friend. Albrecht B¨ ottcher, Israel Gohberg, Bernd Silbermann

Operator Theory: Advances and Applications, Vol. 199, 7–19 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Georg Heinig November 24, 1947 – May 10, 2005 A Personal Memoir and Appreciation Karla Rost

Originally published in Linear Algebra and its Applications 413 (2006), 1–12.

8

K. Rost

On May 10, 2005, Georg Heinig, an excellent mathematician died unexpectedly at the age of 57. He was a world leader in the ﬁeld of structured matrices. As associate editor of the journal Linear Algebra and its Applications since his appointment in 1991 he contributed much to the journal’s success by his valuable and extensive work. In what follows I want to try to capture some aspects of this mathematical life and his personality. Georg was born on November 24, 1947, in the small town of Zschopau in the Ore Mountains (Erzgebirge) in East Germany. From 1954 to 1964 he attended the elementary school there. Because of his good performance he was admitted to the elite school of Karl-Marx-Stadt (now Chemnitz) University of Technology, where he received his graduation diploma with the grade “very good”. During his time at this school he already showed extraordinary talent for mathematics and natural sciences, and his passion and skills for solving mathematical problems grew. Subsequently he studied mathematics at Karl-Marx-Stadt (now Chemnitz) University of Technology. He wrote his diploma thesis under the supervision of Siegfried Pr¨ ossdorf [SP] on some properties of normally solvable operators in Banach spaces. Some years ago S. Pr¨ ossdorf told me that he liked to recall the time he spent with the gifted and creative student Georg. In the summer of 1970 Georg received the best possible grade for the defense of his diploma thesis. He received a scholarship to study for the Ph.D. abroad and he decided to go to the State University of Moldavia at Kishinev from 1971 to 1974. There he worked on his PhD thesis on the subject of Wiener-Hopf block operators and singular integral operators under the supervision of Israel Gohberg [IG] who even then was internationally well known and respected. Here is a copy of the authenticated Russian document certifying Georg’s degree as “candidate of sciences”, which is the equivalent to a PhD:

A well-known, important and often cited work with I. Gohberg from this time is the paper [111]. With great admiration and deep gratitude Georg always considered I. Gohberg as his scientiﬁc father ([IG], page 63). By this time Georg was cast irretrievably into the realm of matrix theory, in particular the theory of structured matrices. His commitment to this ﬁeld over three decades has beneﬁted several scientiﬁc grandchildren of Israel Gohberg’s of whom I am one. Georg’s connection to I. Gohberg has never ceased. From 1993

Georg Heinig – A Personal Memoir and Appreciation

9

on, he was a member of the editorial board of the journal Integral Equations and Operator Theory. I became acquainted with Georg when he returned to Karl-Marx-Stadt in late 1974, where he worked at the Department of Mathematics, ﬁrst in the group chaired by S. Pr¨ ossdorf and then (after Pr¨ ossdorf’s leave to Berlin 1975) in B. Silbermann’s group [BS]. There he found extremely good conditions. Pr¨ ossdorf and Silbermann considered him as an equal partner, and hence he could pursue his inclinations in research unhampered and even build a small research group. In all honesty, I have to admit that I was not euphoric at my ﬁrst encounters with Georg. He was very young, with no experience as a supervisor, and in addition, he appeared to me as too self-oriented. It was his ability to awake my interest in the topics he proposed, which in 1975 led me to decide to write my diploma thesis under his supervision despite my initial hesitations. In fact he turned out to be an extraordinary supervisor, and I soon became aware that starting my scientiﬁc career with him was a lucky decision. In later years we managed better and better to get attuned to each other, and consequently I wrote a large part of my dissertation on the method of UV-reduction for inverting structured matrices under his supervision in 1980. Meanwhile 30 years of fruitful and intense joint work have passed. One joint monograph and almost 40 papers in journals testify to this. In 1979 Georg defended his habilitation thesis (of an imposing length of 287 pages) on the spectral theory of operator bundles and the algebraic theory of ﬁnite Toeplitz matrices with excellence. Georg was very optimistic and in love with life. I very much miss his cheerful and bright laughter. Certainly his stay in Kishinev intensiﬁed his wanderlust and his curiosity for other countries. Despite the travel restrictions for citizens of the G.D.R., the former socialist part of Germany, there was scientiﬁc cooperation with Syria and Ethiopia, and Georg was oﬀered a research and working visit at Aleppo University in Syria in 1982. During a longer stay from 1985–87 at Addis Ababa University in Ethiopia, he was accompanied by his wife Gerti and by his two children Peter (born 1974) and Susanne (born 1977). Later on, both these stays certainly helped him to settle down in Kuwait. Georg was well established at Karl-Marx-Stadt (now Chemnitz) University of Technology. He was a respected and highly recognized colleague with outstanding achievements in research and teaching. Thus Georg was appointed as a full professor for numerical mathematics at Karl Marx University of Leipzig. Since the late seventies his international recognition has grown enormously, which is, for example, reﬂected by the interest of the Birkh¨auser publishing house in the joint monograph [85, 88], which was originally intended to be published by the Akademie-Verlag only. The “Wende” in the fall of 1989 was an incisive break and turning point in the life of many people in East Germany, and thus also for Georg. All scientists working at universities were formally dismissed and had to apply for a position anew. An important criterion for a refusal of such an application was the political

10

K. Rost

proximity to the old socialist system. Due to this, in 1993 Georg went to Kuwait University, where he worked as a professor for more than 10 years. He died of a heart attack on May 10, 2005, in his apartment in Kuwait.

During this long period in Kuwait he continuously maintained scientiﬁc and personal contacts with his friends and former colleagues from Chemnitz, including Albrecht B¨ ottcher, Bernd Silbermann, Steﬀen Roch, and myself. In May 1998 we all had the opportunity to participate in the International Conference on Fourier Analysis and Applications in Kuwait. Georg had an especially high admiration for Albrecht B¨ ottcher and was therefore very glad that Albrecht agreed to enter the scientiﬁc committee and the editorial board of the proceedings of this conference [33]. In the course of that conference we convinced ourselves with great pleasure of the respect in which Georg was held by his colleagues and students in Kuwait. Thus he found very good friends and supporters in his Kuwaiti colleagues Fadhel Al-Musallam and Mansour Al-Zanaidi as well as in his colleague Christian Grossmann from Dresden University of Technology, who stayed in Kuwait from 1992 to 1998. One of the highlights of his life occurred in 2002, when the Amir of Kuwait distinguished him as the Researcher of the Year. Since 2004 he was also a member of the editorial board of the Kuwait Journal of Science and Engineering. My mathematical knowledge and my ability to tackle problems have beneﬁted immensely from Georg. He had an extraordinary gift to explain complicated things in simple terms. This was also appreciated by his students. His lectures and scientiﬁc talks were very sought after and well attended. The aesthetic component is well to the fore in his work. He mastered with equal facility problems of extreme generality and abstraction as well as down-to-earth questions.

Georg Heinig – A Personal Memoir and Appreciation

11

Georg is the author and coauthor of more than 100 scientiﬁc publications. He always made high demands on himself and on his coauthors regarding not only mathematical originality and exactness but also regarding clear and short exposition. His main research interests are – – – – –

structured matrices: algebraic theory and fast algorithms, interpolation problems, operator theory and integral equations, numerical methods for convolution equations, applications in systems and control theory and signal processing.

In each of these topics he achieved essential contributions which is impressively shown by his list of publications. In my opinion, especially the importance of his results concerning the algebraic theory of structured matrices are striking and imposing. In particular, in our joint paper [74] we show that inverses of matrices which are the sum of a Toeplitz and a Hankel matrix possess a Bezoutian structure as inverses of Hankel or Toeplitz matrices do separately. On the basis of this structure, for example, matrix representations can be found and fast algorithms can be designed. Thus a breakthrough for the class of Toeplitz-plus-Hankel matrices was achieved. Moreover, Georg’s observation of the kernel structure of (block) Toeplitz and Toeplitz-plus-Hankel matrices turns out to be a suitable key to develop algorithms without additional assumptions. His ideas how to connect the structure of a matrix with its additional symmetries lead to more eﬃcient inversion and solution algorithms as well as a new kind of factorization. But also his contributions to Toeplitz least square problems, to transformation techniques for Toeplitz and Toeplitz-plus-Hankel matrices leveraged the research in these ﬁelds. I am proud that he has completed (and still wanted to complete) many mathematical projects with me. For many years we dreamt about writing a neat textbook on structured matrices for graduate students, ranging from the basics for beginners up to recent developments. One year ago we really started writing the ﬁrst chapters of this book. Except for some interruptions, when we were very busy with teaching, the collaboration was therefore especially intense, partly culminating in a dozen of emails per day. I received two emails from him on May 10, 2005. I then did not know that they were his last! Georg’s work leaves behind a trail that points to directions for future research. His early death leaves a loss from which we cannot recover, for it is tragic how many plans and original ideas have passed away with him. Colleagues like me are left behind in shock and ask themselves how we can at least partially close the gap that he has left. In such situations the persistent optimist Georg used to say: “Lamenting does not help. Things are as they are. Let us conﬁdently continue to work. This helps!”

12

K. Rost

References [BS] Toeplitz Matrices and Singular Integral Equations: The Bernd Silbermann Anniversary Volume (Pobershau, 2001), eds. A. B¨ottcher, I. Gohberg, P. Junghanns, Operator Theory: Advances and Applications, Vol. 135, Birkh¨auser, Basel, 2002. [SP] Problems and Methods in Mathematical Physics: The Siegfried Pr¨ossdorf Memorial Vol. (Proceed. of the 11 TMP Chemnitz, 1999), eds. J. Elschner, I. Gohberg, B. Silbermann, Operator Theory: Advances and Applications, Vol. 121, Birkh¨auser, Basel, 2001. [IG] The Gohberg Anniversary Collection, Vol. I (Calgary, 1988), eds. H. Dym, S. Goldberg, M.A. Kaashoek, P. Lancaster, Operator Theory: Advances and Applications, Vol. 40, Birkh¨ auser, Basel, 1989.

List of Georg Heinig’s (refereed) publications (chronologically ordered, including one monograph [85, 88], two edited proceedings [14, 33], and one book translation [30]) [1] G. Heinig, K. Rost, Split algorithms for centrosymmetric Toeplitz-plus-Hankel matrices with arbitrary rank proﬁle, 129–146, Oper. Theory Adv. Appl., 171, Birkh¨ auser, Basel, 2007. [2] G. Heinig, K. Rost, Schur-type algorithms for the solution of Hermitian Toeplitz systems via factorization, 233–252, Oper. Theory Adv. Appl., 160, Birkh¨auser, Basel, 2005. [3] G. Codevico, G. Heinig, M. Van Barel, A superfast solver for real symmetric Toeplitz systems using real trigonometric transformations. Numer. Linear Algebra Appl. 12 (2005), 699–713. [4] G. Heinig, K. Rost, Fast “split” algorithms for Toeplitz and Toeplitz-plus-Hankel matrices with arbitrary rank proﬁle. Proceedings of the International Conference on Mathematics and its Applications (ICMA 2004), 285–312, Kuwait, 2005. [5] G. Heinig, K. Rost, Split algorithms for symmetric Toeplitz matrices with arbitrary rank proﬁle. Numer. Linear Algebra Appl. 12 (2005), no. 2-3, 141–151. [6] G. Heinig, K. Rost, Split algorithms for Hermitian Toeplitz matrices with arbitrary rank proﬁle. Linear Algebra Appl. 392 (2004), 235–253. [7] G. Heinig, Fast algorithms for Toeplitz least squares problems. Current trends in operator theory and its applications. 167–197, Oper. Theory Adv. Appl., 149, Birkh¨ auser, Basel, 2004. [8] G. Heinig, K. Rost, Split algorithms for skewsymmetric Toeplitz matrices with arbitrary rank proﬁle. Theoret. Comput. Sci. 315 (2004), no. 2-3, 453–468. [9] G. Heinig, K. Rost, New fast algorithms for Toeplitz-plus-Hankel matrices. SIAM J. Matrix Anal. Appl. 25 (2003), no. 3, 842–857. [10] G. Heinig, K. Rost, Fast algorithms for centrosymmetric and centro-skewsymmetric Toeplitz-plus-Hankel matrices. International Conference on Numerical Algorithms, Vol. I (Marrakesh, 2001), Numer. Algorithms 33 (2003), no. 1-4, 305–317.

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[11] G. Heinig, Inversion of Toeplitz-plus-Hankel matrices with arbitrary rank proﬁle. Fast algorithms for structured matrices: theory and applications (South Hadley, MA, 2001), 75–89, Contemp. Math., 323, Amer. Math. Soc., Providence, RI, 2003. [12] M. Van Barel, G. Heinig, P. Kravanja, A superfast method for solving Toeplitz linear least squares problems. Special issue on structured matrices: analysis, algorithms and applications (Cortona, 2000), Linear Algebra Appl. 366 (2003), 441–457. [13] G. Heinig, K. Rost, Centrosymmetric and centro-skewsymmetric Toeplitz-plusHankel matrices and Bezoutians. Special issue on structured matrices: analysis, algorithms and applications (Cortona, 2000), Linear Algebra Appl. 366 (2003), 257–281. [14] Special issue on structured matrices: analysis, algorithms and applications. Papers from the workshop held in Cortona, September 21–28, 2000. eds. D. Bini, G. Heinig, E. Tyrtyshnikov. Linear Algebra Appl. 366 (2003). Elsevier Science B.V., Amsterdam, 2003. [15] G. Heinig, K. Rost, Fast algorithms for skewsymmetric Toeplitz matrices. Toeplitz matrices and singular integral equations (Pobershau, 2001), 193–208, Oper. Theory Adv. Appl., 135, Birkh¨ auser, Basel, 2002. [16] G. Heinig, On the reconstruction of Toeplitz matrix inverses from columns. Linear Algebra Appl. 350 (2002), 199–212. [17] G. Heinig, K. Rost, Centro-symmetric and centro-skewsymmetric Toeplitz matrices and Bezoutians. Special issue on structured and inﬁnite systems of linear equations. Linear Algebra Appl. 343/344 (2002), 195–209. [18] G. Heinig, Kernel structure of Toeplitz-plus-Hankel matrices. Linear Algebra Appl. 340 (2002), 1–13. [19] G. Heinig, Fast and superfast algorithms for Hankel-like matrices related to orthogonal polynomials. Numerical analysis and its applications (Rousse, 2000), 385–392, Lecture Notes in Comput. Sci., 1988, Springer, Berlin, 2001. [20] M. Van Barel, G. Heinig, P. Kravanja, An algorithm based on orthogonal polynomial vectors for Toeplitz least squares problems. Numerical analysis and its applications (Rousse, 2000), 27–34, Lecture Notes in Comput. Sci., 1988, Springer, Berlin, 2001. [21] M. Van Barel, G. Heinig, P. Kravanja, A stabilized superfast solver for nonsymmetric Toeplitz systems. SIAM J. Matrix Anal. Appl. 23 (2001), no. 2, 494–510. [22] G. Heinig, K. Rost, Eﬃcient inversion formulas for Toeplitz-plus-Hankel matrices using trigonometric transformations. Structured matrices in mathematics, computer science, and engineering, II (Boulder, CO, 1999), 247–264, Contemp. Math., 281, Amer. Math. Soc., Providence, RI, 2001. [23] G. Heinig, Stability of Toeplitz matrix inversion formulas. Structured matrices in mathematics, computer science, and engineering, II (Boulder, CO, 1999), 101–116, Contemp. Math., 281, Amer. Math. Soc., Providence, RI, 2001. [24] G. Heinig, V. Olshevsky, The Schur algorithm for matrices with Hessenberg displacement structure. Structured matrices in mathematics, computer science, and engineering, II (Boulder, CO, 1999), 3–15, Contemp. Math., 281, Amer. Math. Soc., Providence, RI, 2001.

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[25] G. Heinig, Not every matrix is similar to a Toeplitz matrix. Proceedings of the Eighth Conference of the International Linear Algebra Society (Barcelona, 1999). Linear Algebra Appl. 332/334 (2001), 519–531. [26] G. Heinig, Chebyshev-Hankel matrices and the splitting approach for centrosymmetric Toeplitz-plus-Hankel matrices. Linear Algebra Appl. 327 (2001), no. 1-3, 181–196. [27] S. Feldmann, G. Heinig, Partial realization for singular systems in standard form. Linear Algebra Appl. 318 (2000), no. 1-3, 127–144. [28] G. Heinig, K. Rost, Representations of inverses of real Toeplitz-plus-Hankel matrices using trigonometric transformations. Large-scale scientiﬁc computations of engineering and environmental problems, II (Sozopol, 1999), 80–86, Notes Numer. Fluid Mech., 73, Vieweg, Braunschweig, 2000. [29] M. Van Barel, G. Heinig, P. Kravanja, Least squares solution of Toeplitz systems based on orthogonal polynomial vectors. Advanced Signal Processing Algorithms, Architectures, and Implementations X. ed. F.T. Luk, Vol 4116, Proceedings of SPIE (2000), 167-172. [30] V. Maz’ya, S. Nazarov, B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. I. Translated from the German by Georg Heinig and Christian Posthoﬀ. Operator Theory: Advances and Applications, 111. Birkh¨ auser, Basel, 2000. [31] G. Heinig, K. Rost, Hartley transform representations of symmetric Toeplitz matrix inverses with application to fast matrix-vector multiplication. SIAM J. Matrix Anal. Appl. 22 (2000), no. 1, 86–105. [32] G. Heinig, K. Rost, Hartley transform representations of inverses of real Toeplitzplus-Hankel matrices. Proceedings of the International Conference on Fourier Analysis and Applications (Kuwait, 1998). Numer. Funct. Anal. Optim. 21 (2000), no. 1-2, 175–189. [33] Proceedings of the International Conference on Fourier Analysis and Applications. Held at Kuwait University, Kuwait, May 3–6, 1998, Eds. F. Al-Musallam, A. B¨ ottcher, P. Butzer, G. Heinig, Vu Kim Tuan. Numer. Funct. Anal. Optim. 21 (2000), no. 1-2. Marcel Dekker, Inc., Monticello, NY, 2000. [34] G. Heinig, F. Al-Musallam, Hermite’s formula for vector polynomial interpolation with applications to structured matrices. Appl. Anal. 70 (1999), no. 3-4, 331–345. [35] S. Feldmann, G. Heinig, Parametrization of minimal rank block Hankel matrix extensions and minimal partial realizations. Integral Equations Operator Theory 33 (1999), no. 2, 153–171. [36] G. Heinig, K. Rost, DFT representations of Toeplitz-plus-Hankel Bezoutians with application to fast matrix-vector multiplication. ILAS Symposium on Fast Algorithms for Control, Signals and Image Processing (Winnipeg, MB, 1997). Linear Algebra Appl. 284 (1998), no. 1-3, 157–175. [37] G. Heinig, Matrices with higher-order displacement structure. Linear Algebra Appl. 278 (1998), no. 1-3, 295–301. [38] G. Heinig, A. Bojanczyk, Transformation techniques for Toeplitz and Toeplitz-plusHankel matrices. II. Algorithms. Linear Algebra Appl. 278 (1998), no. 1-3, 11–36.

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[39] G. Heinig, Properties of “derived” Hankel matrices. Recent progress in operator theory (Regensburg, 1995), 155–170, Oper. Theory Adv. Appl., 103, Birkh¨ auser, Basel, 1998. [40] G. Heinig, K. Rost, Representations of Toeplitz-plus-Hankel matrices using trigonometric transformations with application to fast matrix-vector multiplication. Proceedings of the Sixth Conference of the International Linear Algebra Society (Chemnitz, 1996). Linear Algebra Appl. 275/276 (1998), 225–248. [41] G. Heinig, F. Al-Musallam, Lagrange’s formula for tangential interpolation with application to structured matrices. Integral Equations Operator Theory 30 (1998), no. 1, 83–100. [42] G. Heinig, Generalized Cauchy-Vandermonde matrices. Linear Algebra Appl. 270 (1998), 45–77. [43] G. Heinig, The group inverse of the transformation S(X) = 3DAX − XB. Linear Algebra Appl. 257 (1997), 321–342. [44] G. Heinig, A. Bojanczyk, Transformation techniques for Toeplitz and Toeplitz-plusHankel matrices. I. Transformations. Proceedings of the Fifth Conference of the International Linear Algebra Society (Atlanta, GA, 1995). Linear Algebra Appl. 254 (1997), 193–226. [45] G. Heinig, L.A. Sakhnovich, I.F. Tidniuk, Paired Cauchy matrices. Linear Algebra Appl. 251 (1997), 189–214. [46] G. Heinig, Solving Toeplitz systems after extension and transformation. Toeplitz matrices: structures, algorithms and applications (Cortona, 1996). Calcolo 33 (1998), no. 1-2, 115–129. [47] S. Feldmann, G. Heinig, On the partial realization problem for singular systems. Proceedings of the 27th Annual Iranian Mathematics Conference (Shiraz, 1996), 79–100, Shiraz Univ., Shiraz, 1996. [48] S. Feldmann, G. Heinig, Vandermonde factorization and canonical representations of block Hankel matrices. Proceedings of the Fourth Conference of the International Linear Algebra Society (ILAS) (Rotterdam, 1994). Linear Algebra Appl. 241/243 (1996), 247–278. [49] G. Heinig, Inversion of generalized Cauchy matrices and other classes of structured matrices. Linear algebra for signal processing (Minneapolis, MN, 1992), 63–81, IMA Vol. Math. Appl., 69, Springer, New York, 1995. [50] G. Heinig, Matrix representations of Bezoutians. Special issue honoring Miroslav Fiedler and Vlastimil Pt´ ak. Linear Algebra Appl. 223/224 (1995), 337–354. [51] G. Heinig, K. Rost, Recursive solution of Cauchy-Vandermonde systems of equations. Linear Algebra Appl. 218 (1995), 59–72. [52] G. Heinig, Generalized inverses of Hankel and Toeplitz mosaic matrices. Linear Algebra Appl. 216 (1995), 43–59. [53] G. Heinig, F. Hellinger, Displacement structure of generalized inverse matrices. Generalized inverses (1993). Linear Algebra Appl. 211 (1994), 67–83. [54] G. Heinig, F. Hellinger, The ﬁnite section method for Moore-Penrose inversion of Toeplitz operators. Integral Equations Operator Theory 19 (1994), no. 4, 419–446.

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[55] G. Heinig, F. Hellinger, Displacement structure of pseudoinverses. Second Conference of the International Linear Algebra Society (ILAS) (Lisbon, 1992). Linear Algebra Appl. 197/198 (1994), 623–649. [56] S. Feldmann, G. Heinig, Uniqueness properties of minimal partial realizations. Linear Algebra Appl. 203/204 (1994), 401–427. [57] G. Heinig, F. Hellinger, Moore-Penrose inversion of square Toeplitz matrices. SIAM J. Matrix Anal. Appl. 15 (1994), no. 2, 418–450. [58] A.W. Boja´ nczyk, G. Heinig, A multi-step algorithm for Hankel matrices. J. Complexity 10 (1994), no. 1, 142–164. [59] G. Heinig, F. Hellinger, On the Bezoutian structure of the Moore-Penrose inverses of Hankel matrices. SIAM J. Matrix Anal. Appl. 14 (1993), no. 3, 629–645. [60] T. Finck, G. Heinig, K. Rost, An inversion formula and fast algorithms for CauchyVandermonde matrices. Linear Algebra Appl. 183 (1993), 179–191. [61] G. Heinig, P. Jankowski, Kernel structure of block Hankel and Toeplitz matrices and partial realization. Linear Algebra Appl. 175 (1992), 1–30. [62] G. Heinig, Inverse problems for Hankel and Toeplitz matrices. Linear Algebra Appl. 165 (1992), 1–23. [63] G. Heinig, Fast algorithms for structured matrices and interpolation problems. Algebraic computing in control (Paris, 1991), 200–211, Lecture Notes in Control and Inform. Sci., 165, Springer, Berlin, 1991. [64] G. Heinig, On structured matrices, generalized Bezoutians and generalized Christoffel-Darboux formulas. Topics in matrix and operator theory (Rotterdam, 1989), 267–281, Oper. Theory Adv. Appl., 50, Birkh¨ auser, Basel, 1991. [65] G. Heinig, Formulas and algorithms for block Hankel matrix inversion and partial realization. Signal processing, scattering and operator theory, and numerical methods (Amsterdam, 1989), 79–90, Progr. Systems Control Theory, 5, Birkh¨auser Boston, Boston, MA, 1990. [66] G. Heinig, P. Jankowski, Parallel and superfast algorithms for Hankel systems of equations. Numer. Math. 58 (1990), no. 1, 109–127. [67] G. Heinig, P. Jankowski, Fast algorithms for the solution of general Toeplitz systems. Wiss. Z. Tech. Univ. Karl-Marx-Stadt 32 (1990), no. 1, 12–17. [68] G. Heinig, K. Rost, Matrices with displacement structure, generalized Bezoutians, and Moebius transformations. The Gohberg anniversary collection, Vol. I (Calgary, AB, 1988), 203–230, Oper. Theory Adv. Appl., 40, Birkh¨ auser, Basel, 1989. [69] G. Heinig, K. Rost, Inversion of matrices with displacement structure. Integral Equations Operator Theory 12 (1989), no. 6, 813–834. [70] G. Heinig, W. Hoppe, K. Rost, Structured matrices in interpolation and approximation problems. Wiss. Z. Tech. Univ. Karl-Marx-Stadt 31 (1989), no. 2, 196–202. [71] G. Heinig, K. Rost, Matrix representations of Toeplitz-plus-Hankel matrix inverses. Linear Algebra Appl. 113 (1989), 65–78. [72] G. Heinig, T. Amdeberhan, On the inverses of Hankel and Toeplitz mosaic matrices. Seminar Analysis (Berlin, 1987/1988), 53–65, Akademie-Verlag, Berlin, 1988. [73] G. Heinig, P. Jankowski, K. Rost, Tikhonov regularisation for block Toeplitz matrices. Wiss. Z. Tech. Univ. Karl-Marx-Stadt 30 (1988), no. 1, 41–45.

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[74] G. Heinig, K. Rost, On the inverses of Toeplitz-plus-Hankel matrices. Linear Algebra Appl. 106 (1988), 39–52. [75] G. Heinig, P. Jankowski, K. Rost, Fast inversion algorithms of Toeplitz-plus-Hankel matrices. Numer. Math. 52 (1988), no. 6, 665–682. [76] G. Heinig, Structure theory and fast inversion of Hankel striped matrices. I. Integral Equations Operator Theory 11 (1988), no. 2, 205–229. [77] G. Heinig, K. Rost, Inversion of generalized Toeplitz-plus-Hankel matrices. Wiss. Z. Tech. Univ. Karl-Marx-Stadt 29 (1987), no. 2, 209–211. [78] G. Heinig, U. Jungnickel, Hankel matrices generated by Markov parameters, Hankel matrix extension, partial realization, and Pad´e-approximation. Operator theory and systems (Amsterdam, 1985), 231–253, Oper. Theory Adv. Appl., 19, Birkh¨auser, Basel, 1986. [79] G. Heinig, U. Jungnickel, Lyapunov equations for companion matrices. Linear Algebra Appl. 76 (1986), 137–147. [80] G. Heinig, U. Jungnickel, Hankel matrices generated by the Markov parameters of rational functions. Linear Algebra Appl. 76 (1986), 121–135. [81] G. Heinig, Partial indices for Toeplitz-like operators. Integral Equations Operator Theory 8 (1985), no. 6, 805–824. [82] G. Heinig, K. Rost, Fast inversion of Toeplitz-plus-Hankel matrices. Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 27 (1985), no. 1, 66–71. [83] G. Heinig, U. Jungnickel, On the Bezoutian and root localization for polynomials. Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 27 (1985), no. 1, 62–65. [84] G. Heinig, B. Silbermann, Factorization of matrix functions in algebras of bounded functions. Spectral theory of linear operators and related topics (Timi¸soara/Herculane, 1983), 157–177, Oper. Theory Adv. Appl., 14, Birkh¨ auser, Basel, 1984. [85] G. Heinig, K. Rost, Algebraic methods for Toeplitz-like matrices and operators. Oper. Theory Adv. Appl., 13, Birkh¨auser, Basel, 1984. [86] G. Heinig, U. Jungnickel, On the Routh-Hurwitz and Schur-Cohn problems for matrix polynomials and generalized Bezoutians. Math. Nachr. 116 (1984), 185–196. [87] G. Heinig, K. Rost, Schnelle Invertierungsalgorithmen f¨ ur einige Klassen von Matrizen. (German) [Fast inversion algorithms for some classes of matrices] Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 26 (1984), no. 2, 235–241. [88] G. Heinig, K. Rost, Algebraic methods for Toeplitz-like matrices and operators. Mathematical Research, 19, Akademie-Verlag, Berlin, 1984. [89] G. Heinig, K. Rost, Invertierung von Toeplitzmatrizen und ihren Verallgemeinerungen. I. Die Methode der U V -Reduktion. (German) [Inversion of Toeplitz matrices and their generalizations. I. The method of U V -reduction] Beitr¨ age Numer. Math. No. 12 (1984), 55–73. [90] G. Heinig, Generalized resultant operators and classiﬁcation of linear operator pencils up to strong equivalence. Functions, series, operators, Vol. I, II (Budapest, 1980), 611–620, Colloq. Math. Soc. J´ anos Bolyai, 35, North-Holland, Amsterdam, 1983. [91] G. Heinig, Inversion of Toeplitz and Hankel matrices with singular sections. Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 25 (1983), no. 3, 326–333.

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[92] G. Heinig, U. Jungnickel, Zur L¨ osung von Matrixgleichungen der Form AX − XB = 3DC. (German) [On the solution of matrix equations of the form AX −XB = 3DC] Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 23 (1981), no. 4, 387–393. [93] G. Heinig, Linearisierung und Realisierung holomorpher Operatorfunktionen. (German) Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 22 (1980), no. 5, 453–459. [94] G. Heinig, K. Rost, Invertierung einiger Klassen von Matrizen und Operatoren. I. Endliche Toeplitzmatrizen und ihre Verallgemeinerungen. (German) [Inversion of some classes of matrices and operators. I. Finite Toeplitz matrices and their generalizations] Wissenschaftliche Informationen [Scientiﬁc Information], 12, Technische Hochschule Karl-Marx-Stadt, Sektion Mathematik, Karl-Marx-Stadt, 1979. [95] G. Heinig, Bezoutiante, Resultante und Spektralverteilungsprobleme f¨ ur Operatorpolynome. (German) Math. Nachr. 91 (1979), 23–43. [96] G. Heinig, Transformationen von Toeplitz- und Hankelmatrizen. (German) Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 21 (1979), no. 7, 859–864. [97] G. Heinig, Invertibility of singular integral operators. (Russian) Soobshch. Akad. Nauk Gruzin. SSR 96 (1979), no. 1, 29–32. ¨ [98] G. Heinig, Uber ein kontinuierliches Analogon der Begleitmatrix eines Polynoms und die Linearisierung einiger Klassen holomorpher Operatorfunktionen. (German) Beitr¨ age Anal. 13 (1979), 111–126. [99] G. Heinig, Verallgemeinerte Resultantenbegriﬀe bei beliebigen Matrixb¨ uscheln. II. Gemischter Resultantenoperator. (German) Wiss. Z. Tech. Hochsch. Karl-MarxStadt 20 (1978), no. 6, 701–703. [100] G. Heinig, Verallgemeinerte Resultantenbegriﬀe bei beliebigen Matrixb¨ uscheln. I. Einseitiger Resultantenoperator. (German) Wiss. Z. Tech. Hochsch. Karl-MarxStadt 20 (1978), no. 6, 693–700. [101] G. Heinig, Endliche Toeplitzmatrizen und zweidimensionale Wiener-Hopf-Opera¨ toren mit homogenem Symbol. II. Uber die normale Auﬂ¨ osbarkeit einer Klasse zweidimensionaler Wiener-Hopf Operatoren. (German) Math. Nachr. 82 (1978), 53–68. [102] G. Heinig, Endliche Toeplitzmatrizen und zweidimensionale Wiener-Hopf-Operatoren mit homogenem Symbol. I. Eigenschaften endlicher Toeplitzmatrizen. (German) Math. Nachr. 82 (1978), 29–52. ¨ [103] G. Heinig, K. Rost, Uber homogene Gleichungen vom Faltungstyp auf einem endlichen Intervall. (German) Demonstratio Math. 10 (1977), no. 3-4, 791–806. ¨ [104] G. Heinig, Uber Block-Hankelmatrizen und den Begriﬀ der Resultante f¨ ur Matrixpolynome. (German) Wiss. Z. Techn. Hochsch. Karl-Marx-Stadt 19 (1977), no. 4, 513–519. [105] G. Heinig, The notion of Bezoutian and of resultant for operator pencils. (Russian) Funkcional. Anal. i Priloˇzen. 11 (1977), no. 3, 94–95. [106] I.C. Gohberg, G. Heinig, The resultant matrix and its generalizations. II. The continual analogue of the resultant operator. (Russian) Acta Math. Acad. Sci. Hungar. 28 (1976), no. 3-4, 189–209. [107] G. Heinig, Periodische Jacobimatrizen im entarteten Fall. (German) Wiss. Z. Techn. Hochsch. Karl-Marx-Stadt 18 (1976), no. 4, 419–423.

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¨ [108] G. Heinig, Uber die Invertierung und das Spektrum von singul¨ aren Integraloperatoren mit Matrixkoeﬃzienten. (German) 5. Tagung u ¨ ber Probleme und Methoden der Mathematischen Physik (Techn. Hochschule Karl-Marx-Stadt, Karl-MarxStadt, 1975), Heft 1, 52–59. Wiss. Schr. Techn. Hochsch. Karl-Marx-Stadt, Techn. Hochsch., Karl-Marx-Stadt, 1975. [109] I.C. Gohberg, G. Heinig, On matrix integral operators on a ﬁnite interval with kernels depending on the diﬀerence of the arguments. (Russian) Rev. Roumaine Math. Pures Appl. 20 (1975), 55–73. [110] I.C. Gohberg, G. Heinig, The resultant matrix and its generalizations. I. The resultant operator for matrix polynomials. (Russian) Acta Sci. Math. (Szeged) 37 (1975), 41–61. [111] I.C. Gohberg, G. Heinig, Inversion of ﬁnite Toeplitz matrices consisting of elements of a noncommutative algebra. (Russian) Rev. Roumaine Math. Pures Appl. 19 (1974), 623–663. [112] G. Heinig, The inversion and the spectrum of matrix Wiener-Hopf operators. (Russian) Mat. Sb. (N.S.) 91(133) (1973), 253–266. [113] G. Heinig, The inversion and the spectrum of matrix-valued singular integral operators. (Russian) Mat. Issled. 8 (1973), no. 3(29), 106–121. [114] I.C. Gohberg, G. Heinig, The inversion of ﬁnite Toeplitz matrices. (Russian) Mat. Issled. 8 (1973), no. 3(29), 151–156. [115] G. Heinig, Inversion of periodic Jacobi matrices. (Russian) Mat. Issled. 8 (1973), no. 1(27), 180–200. Karla Rost Department of Mathematics Chemnitz University of Technology D-09107 Chemnitz, Germany [email protected]

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Georg Heinig – Selected Pictures

With Israel Gohberg

With Vlastimil Ptak

With Eric Kaltofen, Lothar Reichel and Dario Bini

With Israel Gohberg and Karla Rost

With Adam Bojanzcyk

Georg Heinig – Selected Pictures

With Paul Furmann, Uwe Helmke and Volker Mehrmann

With Dario Bini

With Victor Pan and Vadim Olshevsky

With Paul Van Doren

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Georg Heinig – Selected Pictures

In Kuwait

Researcher of the Year award, 2002

Georg Heinig – Selected Pictures

With Gerti

With Gerti

In Ethiopia

With Children and Grandchildren

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Operator Theory: Advances and Applications, Vol. 199, 25–118 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Introduction to Bezoutians Georg Heinig and Karla Rost Mathematics Subject Classiﬁcation (2000). Primary 15-01; Secondary 15A09, 15A23, 65F05. Keywords. Bezoutian, Toeplitz matrix inverses, Hankel matrix inverses, resultant matrix, Toeplitz-plus-Hankel matrix inverses.

Foreword In the present paper we consider classes of matrices the entries of which are in a given ﬁeld F . These matrices have a special structure, they are Bezoutians. Historically, Bezoutians were at ﬁrst introduced in connection with the elimination for the solution of systems of nonlinear algebraic equations and in connection with root localization problems. Only much later their importance for Hankel and Toeplitz matrix inversion became clear. We will introduce three kinds of Bezoutians: Toeplitz Bezoutians, Hankel Bezoutians, and Toeplitz-plus-Hankel Bezoutians. The classes of Toeplitz and Hankel Bezoutians are related to Toeplitz and Hankel matrices in two ways. First, the inverses of Toeplitz and Hankel matrices are Toeplitz and Hankel Bezoutians, respectively. Furthermore, in case where F = C, Hermitian Toeplitz and Hankel Bezoutians are congruent to Toeplitz and Hankel matrices. The class of Toeplitzplus-Hankel Bezoutians includes the inverses of Toeplitz-plus-Hankel matrices. Instead of a summary of the content we will oﬀer the table of contents at the end of this foreword. The present paper is not a usual paper. It originated from the draft of one chapter of a text-book on structured matrices planned by both authors. This textbook for graduate students was intended to range from the basics for beginners up to recent investigations. At the beginning of 2005 the outlines for the ﬁrst three chapters were ready and parts of the text were in an acceptable form when Georg Heinig, the head of this project, unexpectedly died of a heart attack on May 10, 2005. We have lost one of the top experts in the ﬁeld of structured matrices. His death reveals a gap we cannot overcome. This is the tragedy of the planned book and also of the present paper.

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In the last period of our cooperation that had lasted 30 years we mainly worked on the third chapter of the textbook, which was dedicated to Bezoutians, so that I think that this part of the book was perhaps the favorite “child” of Georg. Thus I felt obliged to continue and complete this text to achieve a selfcontained, improved version which can be published separately. I started with a preliminary section to make the presentation more selfcontained. Then I corrected and completed the other sections. Since the Toeplitz-plus-Hankel case was not included, I added main results concerning this case in Sections 11 and 12. Moreover, I ﬁnished, as planned, with exercises – part of which were already discussed with Georg – and then I make some short historical notes and provide hints to literature pursuing and accentuating the topic in diﬀerent directions. I hope I was able to do all these things in such a way that Georg would not be ashamed. In fact it is a hard burden of responsibility for me, in particular, since Georg was an outstanding mathematician with excellent abilities in teaching and in writing papers. A further reason for this paper is that the topic of Bezoutians is very nice, interesting and important with a lot of connections and applications. In the last few years, one can even observe a revival of the interest in Bezoutians, mainly motivated by their importance in many modern ﬁelds such as numerical computing and control theory. Thus it would be very useful to have an introductional paper into this topic, where a lot of properties and relations are systematically collected and explained. I neither intend to quote a huge number of relevant papers nor to mention all corresponding generalizations and applications. What I try and do is to appreciate Georg’s contributions, since his legacy concerning this topic is enormously. In 1971 Georg started his PhD studies at the State University of Moldavia in Kishinev under the supervision of Israel Gohberg. By this time he was irretrievably cast into the realm of structured matrices, in particular of Toeplitz and Hankel matrices as well as Bezoutians. His very early joint papers with I. Gohberg [12], [13] dealt with the inversion of ﬁnite Toeplitz matrices, the papers [14], [15], [16], [17], [18], [19], [20], [21] were dedicated to Bezoutians and resultant matrices mainly for operator-valued polynomials or to continual analogs of resultants and Bezoutians. The main results of these papers were milestones of the research in this ﬁeld. In 1975 Georg waked up my interests in the topic of Toeplitz and Hankel matrices and their inverses. Thus, in 1981, I wrote a large part of my PhD thesis on the method of UV-reduction for inverting structured matrices under his excellent supervision. When we started our work on the book “Algebraic Methods for Toeplitz-like Matrices and Operators” [32], [33] he introduced me, in particular, into the wonderful world of Bezoutians. In this book, Section 2 of Part I is dedicated to Bezoutians and resultant matrices. Some of the results presented there are, of course, also oﬀered here but, as the result of new thoughts about the matter, from another point of view. Moreover, in Subsection 2.2, Part II of [33] we present ﬁrst ideas and results concerning matrices which are the sum of a Toeplitz and a Hankel matrix (brieﬂy

Introduction to Bezoutians

27

T+H matrices). In my opinion, one of our most important joint result is that in 1986 we discovered a Bezoutian structure also for inverses of T+H matrices (see [34]). This was the starting point of a long interesting and fruitful joint work on these special cases of structured matrices In fact, until now I feel a motivation given by Georg to deal with the T+H case (see [48], [64]). Beginning with the joint paper [37] we wrote a number of papers on matrix representations for T+H matrices and their inverses which allow fast matrix-vector multiplication (see, e.g., [39], [38], [40], [42], [43]). Then we dealt with the problem how to connect the Toeplitz or T+H structure of matrices with possibly additional symmetries in order to reduce the number of parameters involved in these formulas or in the corresponding algorithms. Georg’s paper [22] showed that splitting ideas in the spirit of Delsarte and Genin were very promising. The splitting approaches of our joint papers [44], [45], [46] diﬀer from those of [22]. (Note that in [34] the concept of ω-structured matrices was introduced as a generalization of matrices possessing a Toeplitz, Hankel, or T+H structure. This class of and further investigated in [35], [36]. But these considerations are not included in this paper.) I was only one of a large number of Georg’s coauthors and pupils. In particular, Uwe Jungnickel wrote his PhD thesis under Georg’s supervision in 1986. In their joint papers [27], [28] they considered Routh-Hurwitz or Schur-Cohn problems of counting the roots of a given polynomial in a half-plane or in a circle. They investigated Hankel matrices generated by the Markov parameters of rational functions and their importance for partial realization and Pade approximation in [30], [29]. They investigated the connection of Bezoutians and resultant matrices for the solution of matrix equations in [26], [31]. In Section 7 of Part I of [33] some ﬁrst results concerning the Bezoutian structure of generalized inverses are presented. Georg continued this investigation together with his student Frank Hellinger. Their results published in [23], [25], [24] have found perpetual interest by a large community. Since they go beyond the scope of the present paper they are not included. It is not possible to recognize the full extent and importance of Georg’s work concerning Bezoutians. I beg your pardon for all I will forget to mention or I will not appreciate to the due extend. Karla Rost

Contents 1. Preliminaries 1.1 Notation 1.2 Sylvester’s inertia law 1.3 Toeplitz, Hankel, and Toeplitz-plus-Hankel matrices 1.4 Quasi-Toeplitz matrices, quasi-Hankel matrices, quasi T+H matrices 1.5 M¨ obius transformations

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G. Heinig and K. Rost

2. Deﬁnitions and properties for the Hankel and Toeplitz case 2.1 Hankel Bezoutians 2.2 The transformation ∇H 2.3 Uniqueness 2.4 Quasi-H-Bezoutians 2.5 Frobenius-Fischer transformations 2.6 Splitting of H-Bezoutians 2.7 Toeplitz Bezoutians 2.8 The transformation ∇T 2.9 Symmetric and skewsymmetric T -Bezoutians 2.10 Hermitian T -Bezoutians 2.11 Splitting of symmetric T -Bezoutians 2.12 Relations between H- and T -Bezoutians 3. Resultant matrices and matrix representations of Bezoutians 3.1 Kravitsky-Russakovsky formulas 3.2 Matrix representations of Bezoutians 3.3 Bezoutians as Schur complements 4. Inverses of Hankel and Toeplitz matrices 4.1 Inverses of Hankel matrices 4.2 Characterization of fundamental systems 4.3 Christoﬀel-Darboux formula 4.4 Inverses of Toeplitz matrices 4.5 Characterization of fundamental systems 4.6 Inverses of symmetric Toeplitz matrices 4.7 Inverses of skewsymmetric Toeplitz matrices 4.8 Inverses of Hermitian Toeplitz matrices 4.9 Solution of systems 5. Generalized triangular factorizations of Bezoutians 5.1 Division with remainder 5.2 Factorization step for H-Bezoutians 5.3 Euclidian algorithm 5.4 Generalized U L-factorization of H-Bezoutians 5.5 Inertia computation 5.6 Factorization step for T -Bezoutians in the generic case 5.7 LU -factorization of T -Bezoutians 5.8 Non-generic case for T -Bezoutians 5.9 Hermitian T -Bezoutians 6. Bezoutians and companion matrices 6.1 Factorization of the companion 6.2 Functional calculus 6.3 Barnett’s formula 6.4 Barnett’s formula for T -Bezoutians

Introduction to Bezoutians

29

7. Hankel matrices generated by rational functions 7.1 Generating functions of Hankel matrices 7.2 Vandermonde factorization of Hankel matrices 7.3 Real Hankel matrices 7.4 The Cauchy index 7.5 Congruence to H-Bezoutians 7.6 Inverses of H-Bezoutians 7.7 Solving the Bezout equation 8. Toeplitz matrices generated by rational functions 8.1 Generating functions of Toeplitz matrices 8.2 Matrices with symmetry properties 8.3 Vandermonde factorization of nonsingular Toeplitz matrices 8.4 Hermitian Toeplitz matrices 8.5 Signature and Cauchy index 8.6 Congruence to T -Bezoutians 8.7 Inverses of T -Bezoutians 8.8 Relations between Toeplitz and Hankel matrices 9. Vandermonde reduction of Bezoutians 9.1 Non-conﬂuent Hankel case 9.2 Non-conﬂuent Toeplitz case 9.3 Conﬂuent case 10. Root localization problems 10.1 Inertia of polynomials 10.2 Inertia with respect to the real line 10.3 Real roots of real polynomials 10.4 Inertia with respect to the imaginary axis 10.5 Roots on the imaginary axis and positive real roots of real polynomials 10.6 Inertia with respect to the unit circle 10.7 Roots of conjugate-symmetric polynomials 11. Toeplitz-plus-Hankel Bezoutians 11.1 Deﬁnition 11.2 The transformation ∇T +H 11.3 Uniqueness 11.4 Inverses of T + H-Bezoutians 12. Inverses of T + H-matrices 12.1 Fundamental systems 12.2 Inversion of T+H matrices 12.3 Inversion of symmetric T+H matrices 12.4 Inversion of centrosymmetric T+H matrices 12.5 Inversion of centro-skewsymmetric T+H matrices 13. Exercises 14. Notes References

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G. Heinig and K. Rost

1. Preliminaries 1. Notation. Throughout the paper, F will denote an arbitrary ﬁeld. In some sections we restrict ourselves to the case that F = C or R, the ﬁelds of complex or real numbers, respectively. By { e1 , . . . , en } the standard basis of Fn is denoted. Furthermore, 0k will stand for a zero vector of length k. If there is no danger of misunderstanding we will omit the subscript k. As usual, an element of the vector space Fn will be identiﬁed with the corresponding n × 1 (column) matrix. That means ⎡ ⎤ x1 ⎢ ⎥ (xi )ni=1 = (x1 , . . . , xn ) = ⎣ ... ⎦ . xn In all what follows we denote by n (t) , t ∈ F , the vector n (t) = (1, t, t2 , . . . , tn−1 ) .

(1.1)

The Bezoutian concept is convenient introduced in polynomial language. First we introduce “polynomial language” for vectors. For x = (xi )ni=1 ∈ Fn , we consider the polynomial n x(t) = n (t)T x = xk tk−1 ∈ Fn (t) k=1

and call it generating polynomial of x. Polynomial language for matrices means n that we introduce the generating polynomial of an m×n matrix A = [ aij ]m i=1,j=1 ∈ Fm×n as the bivariate polynomial A(t, s) = m (t)T An (s) =

m n

aij ti−1 sj−1 .

i=1 j=1

At several places in this paper we will exploit symmetry properties of matrices. Besides symmetry, skewsymmetry and Hermitian symmetry in the usual sense we also deal with persymmetry and centrosymmetry. To be more precise we introduce some notations. Let Jn be the matrix of the ﬂip operator in Fn mapping (x1 , x2 , . . . , xn ) to (xn , xn−1 , . . . , x1 ) , ⎡ ⎤ 0 1 . ⎦. .. Jn = ⎣ (1.2) 1 0 For a vector x ∈ Fn we denote by xJ the vector Jn x and, in case F = C, by x# the vector Jn x , where x is the vector with the conjugate complex entries, xJ = Jn x

and x# = Jn x .

In polynomial language the latter looks like xJ (t) = x(t−1 )tn−1 ,

x# (t) = x(t−1 )tn−1 .

Introduction to Bezoutians

31

A vector is called symmetric if xJ = x, skewsymmetric if xJ = −x, and conjugate symmetric if x# = x. Let Fn+ (Fn− ) denote the subspace of all symmetric (skewsymmetric) vectors of Fn , and let P± be the matrices 1 P± = (In ± Jn ) . (1.3) 2 These matrices are projections onto Fn± and P+ + P− = In ,

P+ − P− = Jn .

For an n × n matrix A, we denote AJ = Jn AJn

and A# = Jn AJn ,

where A is the matrix with the conjugate complex entries. An n × n matrix A is called persymmetric if AJ = AT . The matrix A is called centrosymmetric if AJ = A. It is called centro-skewsymmetric if AJ = −A and centro-Hermitian if A# = A. 2. Sylvester’s inertia law. Assume that F = C. Let A be an Hermitian n×n matrix. The triple of integers In A = (p+ , p− , p0 ) in which p+ is the number of positive, p− the number of negative, and p0 the number of zero eigenvalues, counting multiplicities, is called the inertia of A. Clearly p+ + p− + p0 = n. The integer sgn A = p+ − p− is called the signature of A. Note that p− + p+ is the rank of A, so that rank and signature of an Hermitian matrix determine its inertia. Two Hermitian n × n matrices A and B are called congruent if there is a nonsingular matrix C such that B = C ∗ AC, where C ∗ denotes the conjugate transpose of C. The following is Sylvester’s inertia law, which will frequently be applied in this paper. Theorem 1.1. Congruent matrices have the same inertia. We will often apply the following version of Sylvester’s inertia law. Corollary 1.2. Let A be an Hermitian m × m matrix and C an m × n matrix with m ≤ n and rank C = m. Then the signatures of A and C ∗ AC coincide. To see that Corollary 1.2 follows from Theorem 1.1 we extend C to a nonsin ∗ A C, where by adding rows at the bottom. Then C ∗ AC = C gular n × n matrix C A is the extension of A by n − m zero columns and zero rows on the right and and thus at the bottom, respectively. This means that C ∗ AC is congruent to A, ∗ = sgn A. sgn C AC = sgn A 3. Toeplitz, Hankel, and Toeplitz-plus-Hankel matrices. Let Tmn be the subspace of Fm×n consisting of all m × n Toeplitz matrices n Tmn (a) = [ ai−j ]m i=1,j=1 ,

m+n−1 a = (ai )m−1 . i=1−n ∈ F

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G. Heinig and K. Rost

The subspace of all m × n Hankel matrices n Hmn (s) = [ si+j−1 ]m i=1,j=1 ,

s = (si )m+n−1 ∈ Fm+n−1 i=1

is denoted by Hmn . The dimension of these subspaces is m+n−1 . The intersection Tmn ∩ Hmn consists of all chess-board matrices, ⎤ ⎡ c b c ··· ⎢ b c b ··· ⎥ ⎥ ⎢ (c, b ∈ F) (1.4) B = ⎢ c b c ··· ⎥ ⎦ ⎣ .. .. .. . . . which form a two-dimensional subspace of Fm×n . The subspace of all m×n matrices Rmn which are the sum of a Toeplitz and a Hankel matrix (brieﬂy T+H matrices) Rmn = Tmn (a) + Hmn (s) is 2(m + n − 2) dimensional. Since for an m × n Hankel matrix Hmn the matrix Hmn Jn is Toeplitz any T+H matrix can be represented in the form (a, b ∈ Fm+n−1 ).

Rmn = Tmn (a) + Tmn (b)Jn

(1.5)

From this another representation is derived involving the projections P± introduced in (1.3), Rmn = Tmn (c)P+ + Tmn (d)P− (1.6) with c = a + b, d = a − b . Obviously, all these representations are not unique (see Exercises 15 and 16). 4. Quasi-Toeplitz matrices, quasi-Hankel matrices, and quasi-T+H matrices. We consider the transformation ∇+ in the space of n × n matrices deﬁned by ∇+ (A) = A − Sn ASnT ,

(1.7)

where Sn is the forward shift in F mapping (x1 , x2 , . . . , xn ) to (0, x1 , . . . , xn−1 ) , ⎤ ⎡ 0 0 0 ⎢ 1 0 0 ⎥ ⎥ ⎢ (1.8) Sn = ⎢ ⎥. . . . . ⎦ ⎣ . . n

0

1

0

It can easily be checked that this transformation is one-to-one. The transformation ∇+ is called shift displacement operator. For a Toeplitz matrix Tn = [ ai−j ]ni,j=1 we have, obviously, ⎤ ⎡ 1 a0

⎢ a1 0 ⎥ ⎥ a0 a−1 . . . a1−n ⎢ ∇+ (Tn ) = ⎢ . , .. ⎥ 1 0 ... 0 ⎣ .. . ⎦ an−1 0 where a0 = 12 a0 . In particular, the rank of ∇+ (Tn ) equals 2, unless Tn is triangular. In the latter case the rank of ∇+ (Tn ) equals 1, unless Tn = O.

Introduction to Bezoutians

33

Notice that if Tn is Hermitian, then ∇+ (Tn ) is also Hermitian, and the signature of ∇+ (Tn ) equals zero, unless Tn is diagonal. (Obviously, Tn diagonal means Tn = a0 In and sgn(∇+ (Tn )) equals the signum of a0 .) Moreover, a matrix A is Toeplitz if and only if the (n − 1) × (n − 1) submatrix in the lower right corner of ∇+ (A) is the zero matrix. An n × n matrix A is called quasi-Toeplitz if rank ∇+ (A) ≤ 2 . Clearly, Toeplitz matrices are also quasi-Toeplitz, but not vice versa. The following proposition gives a complete description of quasi-Toeplitz matrices. Since the proof is an elementary calculation, we leave it to the reader. T − h+ hT− , g± = (gi± )ni=1 , h± = Proposition 1.3. Suppose that ∇+ (A) = g+ g− ± n (hi )i=1 . Then A can be represented as the sum of 2 products of triangular Toeplitz matrices, ⎤⎡ − ⎤ ⎡ + ⎤⎡ − ⎤ ⎡ + g1 . . . gn− h1 h1 . . . h − g1 0 0 n ⎥⎢ ⎥⎢ ⎢ .. ⎥ −⎢ .. .. ⎥ . .. .. .. .. A = ⎣ ... ⎦⎣ ⎦⎣ . . . . . ⎦ ⎣ . . ⎦ + − + + + 0 g1 0 h− gn . . . g1 hn . . . h1 1 (1.9) T Conversely, if A is given by (1.9), then ∇+ (A) = g+ g− − h+ hT− .

Analogously, we consider the transformation ∇+ : Fn −→ Fn deﬁned by ∇+ (A) = Sn A − ASnT .

(1.10)

A matrix A is Hankel if and only if the (n − 1) × (n − 1) submatrix in the lower right corner of ∇+ (A) is the zero matrix. We call a matrix A quasi-Hankel if rank ∇+ (A) ≤ 2 . A similar representation to (1.9) can be obtained. Let Wn be the matrix Wn = Sn + SnT , and let ∇ : Fn −→ Fn be deﬁned by ∇ (A) = AWn − Wn A .

(1.11)

Proposition 1.4. A matrix A is a T+H matrix if and only if the (n − 2) × (n − 2) submatrix in the center of ∇ (A) is the zero matrix. We call a matrix A quasi-T+H if rank ∇ (A) ≤ 4 . T+H matrices are also quasi-T+H, but not vice versa. case 5. M¨ obius transformations. The ﬂip operator Jn introduced in (1.2) is a special

a c of a class of operators which will be described in this subsection. Let φ = b d be a nonsingular 2 × 2 matrix with entries from F. We associate φ with the linear fractional function at + b φ(t) = . ct + d Despite we use the name “function”, φ(t) is understood here in a formal sense, i.e., t is considered as an abstract variable. In the case where F = C, φ(t) can be seen as a function mapping the Riemann sphere onto itself. These linear fractional functions form a group M with respect to composition. This group is isomorphic, modulo multiples of I2 , to the group GL(F2 ) of nonsingular 2 × 2 matrices. The

34

G. Heinig and K. Rost

latter means that if φ = φ1 φ2 , then φ(t) = φ2 (φ1 (t)), and φ(t) = t if and only if φ = αI2 for some α ∈ F. We will make use of the fact that the group GL(F2 ) is generated by matrices of the form

0 1 1 0 a 0 . (1.12) , (c) (a = 0), (b) (a) 1 0 b 1 0 1 For φ ∈ GL(F2 ) and a natural number n, let Kn (φ) denote the operator deﬁned by Kn (φ)x(t) = x(φ(t))(ct + d)n−1 for x(t) ∈ Fn (t). An operator of this form will be called M¨ obius transformation. It is easily checked that Kn (φ) maps Fn (t) into itself and is linear. In the special cases (1.12) we have (a) Kn (φ)x(t) = x(at), (b) Kn (φ)x(t) = x(t + b), (c) Kn (φ)x(t) = x(t−1 )tn−1 . (1.13) The matrix representations of these transformations (called M¨ obius matrices) with respect to the standard basis in Fn (t) are n−1

k k−j (a) Kn (φ) = diag(aj )n−1 , (b) K (φ) = b , (c) Kn (φ) = Jn . n j=0 j j,k=0 (1.14) Furthermore, the following is true. Proposition 1.5. If φ1 , φ2 ∈ GL(F2 ) and φ = φ1 φ2 , then Kn (φ) = Kn (φ1 )Kn (φ2 ). It is suﬃcient to prove the proposition for the special matrices (1.12). We leave this to the reader. According to Proposition 1.5 the M¨ obius transformations are all invertible and form a subgroup of the group of invertible linear operators on Fn . Furthermore, Kn (φ)−1 = Kn (φ−1 ). M¨ obius matrices are mostly used in connection with matrix transformations of the form A → Kn (ψ)T AKn (φ)

(1.15)

for ﬁxed φ, ψ ∈ GL(F ). In the literature special cases of such transformations are called Frobenius-Fischer transformations. We will use this name for all transformations of this form. Some Frobenius-Fischer transformations are mappings inside a class of structured matrices, other build a bridge between diﬀerent classes. We discuss here the situation with Hankel and Toeplitz matrices. Let Hn denote the class of n × n Hankel matrices Hn (s) = [ si+j−1 ]ni,j=1 , where s = (si )2n−1 ∈ F2n−1 . The following proposition describes Frobeniusi=1 Fischer transformations that map Hn into itself. 2

Proposition 1.6. For φ ∈ GL(F2 ) and s ∈ F2n−1 , the equality s) Kn (φ)T Hn (s)Kn (φ) = Hn ( with s = K2n−1 (φ)T s is satisﬁed.

Introduction to Bezoutians

35

Proof. It suﬃces to prove the proposition for the special cases (1.12). The cases (a) and (c) are obvious. Let φ be now of the form (b), Kn (φ)T Hn (s)Kn (φ) = [ gij ]n−1 i,j=0 . Then

i+j j i i j i+j−k−l i + j i+j−r b b gij = sk+l+1 = sr+1 . k l r r=0 k=0 l=0

This implies the assertion. [ ai−j ]ni,j=1 .

The Now we consider besides Hankel also Toeplitz matrices Tn = class of n × n Toeplitz matrices will be denoted by Tn . Obviously, Tn is Toeplitz if and only if Jn Tn is Hankel. Remember that Jn is the special M¨ obius matrix Kn (J2 ). Thus modiﬁcations of Propositions 1.6 can be stated about FrobeniusFischer transformations transforming Toeplitz into Hankel, Hankel into Toeplitz and Toeplitz into Toeplitz matrices. In particular, we have the following. Corollary 1.7. For φ ∈ GL(F2 ), the transformation A → Kn (ψ)T A Kn (φ) maps 2. Tn into Hn if ψ = J2 φ, 1. Hn into Hn if ψ = φ, 3. Hn into Tn if ψ = φJ2 , 4. Tn into Tn if ψ = J2 φJ2 . In the case F = C we are in particular interested in congruence transformations, i.e., transformations that preserve Hermitian symmetry. For this we T ∗ have to check under which condition Kn (ψ) = Kn (φ). In terms of the matrix a c this condition is equivalent to φ= b d 1. a, b, c, d real, 2. a = b , c = d , 3. a = c , b = d , 4. a = d , b = c in the cases of Corollary 1.7. In terms of the linear fractional function φ(t) = at + b this means that ct + d 1. φ(t) maps R to R , 2. φ(t) maps T to R , 3. φ(t) maps R to T , 4. φ(t) maps T to T , where T denotes the unit circle. For transformations with this property the inertia of the matrix remains invariant by Sylvester’s inertia law.

2. Deﬁnitions and properties for the Hankel and Toeplitz case 1. Hankel Bezoutians. Let u(t), v(t) ∈ Fn+1 (t) be two polynomials. The Hankel Bezoutian or brieﬂy H-Bezoutian of u(t) and v(t) is, by deﬁnition, the n×n matrix B = BezH (u, v) with the generating polynomial u(t)v(s) − v(t)u(s) . t−s We will also say that B is the H-Bezoutian of the vectors u and v. It is easily seen that B(t, s) is really a polynomial in t and s. A simple argumentation for this is as follows. We ﬁx s = s0 . Then the numerator is a polynomial in t vanishing at t = s0 . Hence we obtain a polynomial after dividing the numerator by t − s0 . Thus B(t, s) =

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G. Heinig and K. Rost

B(t, s0 ) is a polynomial in t. Analogously B(t0 , s) is a polynomial in s for ﬁxed t = t0 , and so the claim is proved. For example, if u(t) = t − a and v(t) = t − b, then we have B(t, s) =

(t − a)(s − b) − (t − b)(s − a) = a−b. t−s

Thus BezH (u, v) = a − b. As a second example we consider the ﬂip matrix Jn introduced in (1.2). The generating function of Jn is Jn (t) =

tn − s n . t−s

Thus Jn is the H-Bezoutian of the polynomials tn and 1 or, in other words, of the last and ﬁrst unit vectors en+1 and e1 . As a more general example, let us compute the H-Bezoutian Bk of a general polynomial u(t) ∈ Fn+1 (t) and ek (t) = tk−1 , k = 1, . . . , n + 1. Suppose that n+1 u(t) = ui ti−1 . Then we have i=1

Bk (t, s) =

n+1

ui

i=1

=

n+1 i=k+1

ti−1 sk−1 − tk−1 si−1 t−s

ti−k − si−k k−1 k−1 tk−i − sk−i i−1 i−1 ui s − ui . t t s t−s t−s i=1 k−1

In matrix language this means that ⎡ ⎢ ⎢ ⎢ ⎢ BezH (u, ek ) = ⎢ ⎢ ⎢ ⎢ ⎣

−u1

. .. ... O

⎤

−u1 .. .

O

−uk−1 uk+1 .. .

... . ..

un+1

⎥ ⎥ ⎥ ⎥ ⎥ . ⎥ ⎥ ⎥ ⎦

(2.1)

un+1 The case k = 1 is of particular importance. For this reason we introduce the notation ⎡ ⎤ . . . un+1 u2 ⎢ ⎥ . B(u) = BezH (u, e1 ) = ⎣ ... (2.2) ⎦. .. un+1 H-Bezoutians are obviously symmetric matrices. They are skewsymmetric with respect to the arguments, i.e., BezH (u, v) = −BezH (v, u) .

Introduction to Bezoutians

37

Furthermore, BezH (u, v) is linear in each argument. That means that, for c1 , c2 ∈ F, BezH (c1 u1 + c2 u2 , v) = c1 BezH (u1 , v) + c2 BezH (u2 , v) . To present a product rule for H-Bezoutians we need the matrix of the multiplication m operator which is introduced as follows. Let a(t) = ak tk ∈ Fm+1 (t). For n = k=0

1, 2, . . . , deﬁne a linear operator Mn (a) : Fn −→ Fm+n by (Mn (a)x)(t) = a(t)x(t) . to the standard bases is the (m + n) × n ⎤⎫ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎥⎪ ⎪ ⎥⎪ .. ⎪ ⎪ ⎥ . ⎬ ⎥⎪ ⎥ .. m+n. (2.3) . a0 ⎥ ⎥⎪ ⎪ ⎪ a1 ⎥ ⎪ ⎥⎪ ⎪ .. ⎥⎪ .. ⎪ ⎪ ⎦ . ⎪ . ⎪ ⎭ am m n Moreover, we need the following matrix. Let a(t) = ak tk and b(t) = b k tk

The matrix of this operator with respect matrix ⎡ a0 ⎢ a1 a0 ⎢ ⎢ .. ⎢ . a1 ⎢ ⎢ .. Mn (a) = ⎢ am . ⎢ ⎢ a m ⎢ ⎢ ⎣

k=0

k=0

be two given polynomials. Then (x(t), y(t)) → a(t)x(t) + b(t)y(t),

x(t) ∈ Fn (t), y(t) ∈ Fm (t)

is a linear operator from the direct product Fn (t) ⊗ Fm (t) to Fm+n (t). The matrix of this operator with respect to the standard bases is given by [ Mn (a) Mm (b) ] . (Here we identify (x(t), y(t)) with x(t) + tn y(t) .) The transpose of this matrix is called the resultant matrix (or Sylvester matrix) of a(t) and b(t) (or of the vectors a and b) and is denoted by Res(a, b),

Mn (a)T . (2.4) Res(a, b) = Mm (b)T If we assume that am = 0 or bn = 0 then Res(a, b) is nonsingular if and only if a(t) and b(t) are coprime (cf. Exercise 3). Proposition 2.1. Let u, v ∈ Fn+1 , u(t) = u1 (t)u2 (t), v(t) = v1 (t)v2 (t), where ui , vi ∈ Fni +1 (i = 1, 2) and n1 + n2 = n − 1. Then

O BezH (u1 , v1 ) T Res (v2 , u1 ) . BezH (u, v) = Res (u2 , v1 ) O BezH (u2 , v2 ) (2.5) Proof. Let B = BezH (u, v). Then, B(t, s) has the representation u2 (t)

u1 (t)v1 (s) − v1 (t)u1 (s) u2 (t)v2 (s) − v2 (t)u2 (s) v2 (s) + v1 (t) u1 (s) . t−s t−s

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In matrix language this means B = Mn1 (u2 )BezH (u1 , v1 )Mn1 (v2 )T + Mn2 (v1 )BezH (u2 , v2 )Mn2 (u1 )T . From this relation the assertion is immediate.

2. The transformation ∇H . Next we clarify what means for a matrix to be an H-Bezoutian in matrix language. For this we introduce the transformation ∇H n transforming an n × n matrix A = [ aij ]i,j=1 into a (n + 1) × (n + 1) matrix according to ∇H A = [ ai−1,j − ai,j−1 ]n+1 i,j=1 . Here we set aij = 0 if one of the integers i or j is not in the set {1, 2, . . . , n}. We have

∗ ∗ Sn A − ASnT ∗ ∇H A = = , (2.6) ∗ ASn − SnT A ∗ ∗ where Sn is deﬁned in (1.8). Comparing the coeﬃcients it is easy to verify that (∇H A)(t, s) = (t − s)A(t, s) . Hence the Bezoutians B = BezH (u, v) can be characterized with the help of ∇H by

0 1 [ u v ]T . (2.7) ∇H B = [ u v ] −1 0 In particular, the rank of ∇BH is equal to 2, unless u and v are linearly dependent. In the latter case the H-Bezoutian is the zero matrix. The representation (2.6) shows that the transformation ∇+ introduced in (1.10) is a restriction of ∇H . Thus, H-Bezoutians are quasi-Hankel matrices. 3. Uniqueness. Diﬀerent pairs of polynomials may produce the same H-Bezoutian. However, from (2.7) one can conclude that if BezH (u, v) = BezH (u1 , v1 ) = O, then span{u, v} = span{u1 , v1 }. In the latter case there is a nonsingular 2 × 2 matrix ϕ such that (2.8) [ u1 v1 ] = [ u v ] ϕ . Lemma 2.2. Let u, v, u1 , v1 ∈ Fn+1 related via (2.8). Then BezH (u1 , v1 ) = (det ϕ) BezH (u, v) .

a c , B1 = BezH (u1 , v1 ). Then Proof. Suppose that ϕ = b d (t − s)B1 (t, s) = (au(t) + bv(t))(cu(s) + dv(s)) − (cu(t) + dv(t))(au(s) + bv(s)) = (ad − bc)(u(t)v(s) − v(t)u(s)) , which proves the lemma.

Corollary 2.3. The H-Bezoutians BezH (u, v) = O and BezH (u1 , v1 ) coincide if and only if the vectors u, v and u1 , v1 are related via (2.8) with det ϕ = 1.

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From Corollary 2.3 we can conclude that the H-Bezoutian BezH (u, v) is equal ) in which the last coeﬃcient of v vanishes, i.e., v (t) ∈ to a H-Bezoutian BezH ( u, v Fn (t). 4. Quasi-H-Bezoutians. A matrix B is called quasi-H-Bezoutian if rank ∇H B ≤ 2. We give a general representation of quasi-H-Bezoutians that is also important for H-Bezoutians. Proposition 2.4. A quasi-H-Bezoutian B = O of order n admits a representation B = Mr (p) BezH (u, v) Mr (q)T , where u(t), v(t) ∈ F

r+1

(2.9)

(t) are coprime and r ≤ n . Here Mr ( · ) is deﬁned in (2.3).

Proof. For B is a quasi-H-Bezoutian, there exist a, b, c, d ∈ Fn+1 such that (t − s)B(t, s) = a(t)d(s) − b(t)c(s) . Since for t = s the left-hand side vanishes, we have a(t)d(t) = b(t)c(t). Let p(t) be the greatest common divisor of a(t) and b(t) and q(t) the greatest common divisor of c(t) and d(t). Then a(t) = p(t)u(t) and b(t) = p(t)v(t) for some coprime u(t), v(t) ∈ Fr+1 (t) (r ≤ n). Furthermore, c(t) = q(t)u1 (t) and d(t) = q(t)v1 (t) for some coprime u1 (t), v1 (t) ∈ Fr1 +1 (t) (r1 ≤ n). Since a(t) u(t) c(t) u1 (t) = = = , b(t) v(t) d(t) v1 (t) we conclude that, for some γ = 0, u1 = γu, v1 = γv, and r = r1 . Now we have a(t)d(s) − b(t)c(s) = γ p(t)(u(t)v(s) − v(t)u(s))q(s) . We can replace γ p by p. Now it remains to translate this into matrix language to obtain (2.9). The matrix on the right-hand side of (2.9) has rank r at most. Hence if r < n, then B is singular. This leads to the following somehow surprising conclusion. Corollary 2.5. Any nonsingular quasi-H-Bezoutian is an H-Bezoutian of two coprime polynomials. Later we will show that, vice versa, the H-Bezoutian of two coprime polynomials is nonsingular (cf. Corollary 3.4). If the quasi-H-Bezoutian is symmetric, then in (2.9) we must have q = p, since the middle factor is symmetric. This implies the following. Corollary 2.6. Any symmetric quasi-H-Bezoutian is an H-Bezoutian B = BezH (a, b) . In particular, (2.9) can be written in the form B = Mr (p) BezH (u, v) Mr (p)T , where p(t) is the greatest common divisor of a(t) and b(t) .

(2.10)

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5. Frobenius-Fischer transformations. We show now that Frobenius-Fischer transformations introduced in Section 1 transform the class of H-Bezoutians into itself. In particular, the following result is the Bezoutian counterpart of Proposition 1.6. Theorem 2.7. For any ϕ ∈ GL(F2 ), the transformation B → Kn (ϕ)BKn (ϕ)T maps H-Bezoutians into H-Bezoutians. Moreover 1 ) , BezH ( u, v Kn (ϕ)BezH (u, v)Kn (ϕ)T = det ϕ = Kn+1 (ϕ)u and v = Kn+1 (ϕ)v. where u

(2.11)

= BezH ( ). It is suﬃcient Proof. Let B = BezH (u, v) and B u, v to prove the

a 0 2 , then theorem for the matrices (1.12) that generate GL(F ). If ϕ = 0 1 s) = a u(at)v(as) − v(at)u(as) = a B(at, as) = a (Kn (ϕ)BKn (ϕ)T )(t, s) , B(t, at − as

1 0 , then which is equivalent to (2.11). If ϕ = b 1 s) = B(t, =

u(t + b)v(s + b) − v(t + b)u(s + b) = B(t + b, s + b) t−s (Kn (ϕ)BKn (ϕ)T )(t, s) ,

which is equivalent to (2.11). Finally, let ϕ = J2 . Then s) = B(t, =

u(t−1 )tn v(s−1 )sn − v(t−1 )tn u(s−1 )sn = −B(t−1 , s−1 )(ts)n−1 t−s −B J (t, s) ,

which is again equivalent to (2.11).

6. Splitting of H-Bezoutians. In some applications, like stability tests for real polynomials, u(t) and v(t) have the special form u(t) = a(t2 ) and v(t) = t b(t2 ).

(2.12)

That means u(t) has only even powers and v(t) only odd powers. In this case we have for the generating polynomial of B = BezH (u, v) after multiplying numerator and denominator by (t + s) B(t, s) = =

ts(a(t2 )b(s2 ) − b(t2 )a(s2 )) + a(t2 )s2 b(s2 ) − t2 b(t2 )a(s2 ) t2 − s2 2 2 2 2 tsB1 (t , s ) + B0 (t , s ) ,

where B1 = BezH (a, b) and B0 = BezH (a, tb). To translate this into matrix language we introduce the matrix Σn of the even-odd shuﬄe operator: Σn (xi )ni=1 = (x1 , x3 , . . . , x2 , x4 , . . . ) .

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41

Proposition 2.8. Let u(t) ∈ Fn (t) and v(t) ∈ Fn (t) be given by (2.12). Then

BezH (a, t b) O ΣTn BezH (u, v)Σn = . O BezH (a, b) 7. Toeplitz Bezoutians. We introduce the Toeplitz analogue of the H-Bezoutian. The Toeplitz Bezoutian or brieﬂy T-Bezoutian of the two polynomials u(t) ∈ Fn+1 (t) and v(t) ∈ Fn+1 (t) is, by deﬁnition, the matrix B = BezT (u, v) with the generating polynomial u(t)vJ (s) − v(t)uJ (s) . 1 − ts Like for H-Bezoutians, it is easily checked that B(t, s) is really a polynomial in t and s. If, for example, the polynomials u(t) = t − a and v(t) = t − b of Fn+1 (t) are given, then for n = 1 B(t, s) =

(t − a)(1 − bs) − (t − b)(1 − as) = b−a. 1 − ts Hence BezT (u, v) = b − a . But in case n > 1 we have

O b−a BezT (u, v) = . O O B(t, s) =

We state that the deﬁnition of a T-Bezoutian of two polynomials depends, in contrast to the H-Bezoutian, essentially on the integer n. That means if we consider u(t) and v(t) as elements of FN +1 (t) for N > n, then we will have a diﬀerent TBezoutian. Indeed, let BN denote the T-Bezoutian in this sense. Then we obtain u(t)v(s−1 )sN − v(t)u(s−1 )sN = B(t, s)sN −n , 1 − ts where B is the T-Bezoutian of u and v in the original sense. Thus, BN is of the form

O B . BN = O O If t = 0 is a common zero of u(t) and v(t), u(t) = tr u0 (t), v(t) = tr v0 (t) (r > 0) , then B is of the form

O O B= B0 O where B0 is the (n − r) × (n − r) T-Bezoutian of u0 and v0 . As an example, we compute the T-Bezoutian of a polynomial and a power of t. Let B(k) = BezT (u, ek ) and u = (ui )n+1 i=1 . Then BN (t, s) =

B(k) (t, s) =

n+1 i=1

=

k−1 i=1

ui

ti−1 sn−k+1 − tk−1 sn−i+1 1 − ts

ui ti−1 sn−k+1

n+1 1 − (ts)k−i (ts)i−k − 1 + . ui tk−1 sn−i+1 1 − ts 1 − ts i=k+1

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We take into account that in matrix form

1 − (ts)k is the generating polynomial of Ik and obtain 1 − ts

⎡

⎢ ⎢ ⎢ ⎢ BezT (u, ek ) = ⎢ ⎢ −un+1 ⎢ ⎢ ⎣

u1 .. .

O ... .. .

−uk+1 .. .

uk−1

⎤ ..

. ... O

⎥ ⎥ ⎥ u1 ⎥ ⎥ . ⎥ ⎥ ⎥ ⎦

(2.13)

−un+1 For the special cases k = 1 and k = n + 1 we introduce the notations ⎡ ⎡ ⎤ ⎤ un+1 . . . u1 u2 ⎢ ⎥ .. ⎥ , B (u) = ⎢ .. .. .. B+ (u) = − ⎣ ⎣ . ⎦. − . . . ⎦ un . . . u1 un+1

(2.14)

Obviously, the T-Bezoutian is, like the H-Bezoutian, linear in its arguments. Furthermore, it is skewsymmetric with respect to the arguments, i.e., BezT (u, v) = −BezT (v, u) . Moreover, we have BezT (u, v)T = −BezT (uJ , vJ )

(2.15)

and BezT (u, v)J (t, s)

=

u(t−1 )v(s)s−n − v(t−1 )u(s)s−n n−1 n−1 t s 1 − t−1 s−1

=

u(t−1 )tn v(s) − v(t−1 )tn u(s) , ts − 1

which means that BezT (u, v)J = −BezT (uJ , vJ ) .

(2.16)

From (2.15) and (2.16) we conclude that BezT (u, v)T = BezT (u, v)J . Hence T-Bezoutians are, like Toeplitz matrices, persymmetric. The discussion about uniqueness of H-Bezoutians in Section 2.1 can be immediately transferred to T-Bezoutians. In fact, there is a Toeplitz analogue of Lemma 2.2. Lemma 2.9. If [ u1 v1 ] = [ u v ] ϕ for some 2 × 2 matrix ϕ, then BezT (u1 , v1 ) = (det ϕ) BezT (u, v) . Hence the following is true.

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43

Corollary 2.10. The T-Bezoutians BezT (u, v) = O and BezT (u1 , v1 ) coincide if and only if [ u1 v1 ] = [ u v ] ϕ for some matrix ϕ with det ϕ = 1 . 8. The transformation ∇T . The Toeplitz analogue of the transformation ∇H is the transformation ∇T transforming an n × n matrix A = [ aij ]ni,j=1 into a (n + 1) × (n + 1) matrix according to ∇T A = [ aij − ai−1,j−1 ]n+1 i,j=1 . Here we set aij = 0 if one of the integers i or j is not in the set {1, 2, . . . , n}. Obviously,

A − Sn ASnT ∗ ∗ ∗ ∇T A = = . (2.17) ∗ SnT ASn − A ∗ ∗ In polynomial language the transformation ∇T is given by (∇T A)(t, s) = (1 − ts)A(t, s) . That means the T-Bezoutian B = BezT (u, v) is characterized by

0 1 [ uJ vJ ]T . ∇T B = [ u v ] −1 0 Taking into account (2.16) we observe that (∇T B)J = −∇T B J . The representation (2.17) shows that the transformation ∇+ introduced in (1.7) is a restriction of ∇T . In particular, we conclude that T-Bezoutians are quasi-Toeplitz matrices. Furthermore, if B is a T-Bezoutian, then B J is also a quasi-Toeplitz matrix. 9. Symmetric and skewsymmetric T-Bezoutians. We discuss now how symmetric and skewsymmetric T-Bezoutians can be characterized. First we observe that (2.16) implies that BezT (u, v) is symmetric if one of the vectors u or v is symmetric and the other one is skewsymmetric. Furthermore, BezT (u, v) is skewsymmetric if both vectors u and v are symmetric or both are skewsymmetric. We show that the converse is also true. For simplicity of notation we write B(u, v) instead of BezT (u, v). Let u, v ∈ Fn+1 be any vectors, u = u+ +u− and v = v+ +v− , where u+ , v+ are symmetric and u− , v− are skewsymmetric. Then B(u, v) = B+ + B− , where B+ = B(u+ , v− ) + B(u− , v+ ) ,

B− = B(u+ , v+ ) + B(u− , v− ) ,

B+ is symmetric, and B− is skewsymmetric. Suppose that B = B(u, v) is symmetric. Then B− = O. Hence B(u+ , v+ ) = B(v− , u− ) . Since the vectors u+ and v+ cannot be linear combinations of u− and v− from Corollary 2.10 it becomes clear that B(u+ , v+ ) = B(v− , u− ) = O .

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Thus v± = α± u± for some α± ∈ F or B(u, v) = O. We conclude that B = B+ = B((α− − α+ )u+ , u− ) . That means that B is the T-Bezoutian of a symmetric and a skewsymmetric vector. Suppose now that B = B(u, v) = O is skewsymmetric. Then B+ = O. Hence B(u+ , v− ) = B(v+ , u− ) . From Corollary 2.10 and the symmetry properties of the vectors we conclude that either {u+ , v+ } as well as {u− , v− } are linearly dependent or B(u+ , v− ) = B(v+ , u− ) = O . In the former case we would have B− = O, so we have the latter case. Using again the symmetry properties of the vectors we ﬁnd that either u− = v− = 0 or u+ = v+ = 0. That means that B is the Bezoutian of two symmetric or two skewsymmetric vectors. Let us summarize. Proposition 2.11. A T-Bezoutian is symmetric if and only if it is the T-Bezoutian of a symmetric and a skewsymmetric vector. A T-Bezoutian is skewsymmetric if and only if it is the T-Bezoutian of two symmetric vectors or two skewsymmetric vectors. Note that the T-Bezoutian B(u, v) of two skewsymmetric vectors cannot be nonsingular. In fact, in this case we have u(1) = v(1) = 0 such that u(t) = (t − 1)u1 (t) and v(t) = (t − 1)v1 (t). Then u1 and v1 are symmetric, and as in Proposition 2.4 we obtain B(u, v) = Mn−1 (t − 1)BezT (u1 , v1 )Mn−1 (t − 1)T . Thus B(u, v) has rank n − 1 at most. There is an alternative representation for symmetric T-Bezoutians, which has no skewsymmetric counterpart. Suppose that B = B(u+ , u− ). We set v = − 12 u+ + u− . Then B(v, vJ ) = B . On the other hand, B(v, vJ ) is symmetric for any vector v ∈ Fn+1 . Thus the following is true. Corollary 2.12. A T-Bezoutian B is symmetric if and only if it can be represented in the form B = BezT (v, vJ ) for some v ∈ Fn+1 . 10. Hermitian T-Bezoutians. Now we characterize Hermitian T-Bezoutians. Suppose that u+ , u− ∈ Cn+1 are conjugate-symmetric. Then we conclude from (2.15) that the matrix iB(u+ , u− ) is Hermitian. Conversely, let B = B(u, v) be Hermitian, u = u+ + iu− and v = v+ + iv− , where u± , v± are conjugate-symmetric. Then B(u, v) = B+ + iB− , where B+ = i(B(u+ , v− ) + B(u− , v+ )) ,

B− = i(B(u− , v− ) − B(u+ , v+ )) .

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45

The matrices B+ and iB− are Hermitian. Since B = B(u, v) is assumed to be Hermitian, we have B− = O, which means B(u+ , v+ ) = B(u− , v− ) . Using Corollary 2.10 we conclude that

[ u− v− ] = [ u+ v+ ]

a b

c d

for some a, b, c, d with ad − bc = 1. Since all vectors under consideration are conjugate-symmetric, these numbers must be real. We obtain after elementary calculations B(u, v) = iB(u+ , −(a + d)v+ ) = B(u+ , −(a + d)iv+ ) . Thus, B is the Bezoutian of a conjugate-symmetric and a conjugate-skewsymmetric vector. Let us summarize. Proposition 2.13. A T-Bezoutian B is Hermitian if and only if it is of the form B = iBezT (u+ , u− ) for conjugate-symmetric vectors u+ and u− . As for symmetric T-Bezoutians, we have an alternative form. Suppose that B = iB(u+ , u− ) and set v = − 12 u+ + iu− . Then B(v, v# ) = B . Since, on the other hand, the matrix B(v, v# ) is Hermitian for any vector v ∈ Fn+1 , which is easily checked, the following is true. Corollary 2.14. A T-Bezoutian B is Hermitian if and only if it can be represented in the form B = BezT (v, v# ) for some v ∈ Cn+1 . 11. Splitting of symmetric T-Bezoutians. It was mentioned in Section 2.7 that T-Bezoutians are persymmetric. Hence a symmetric T-Bezoutian B is also centrosymmetric. That means that the subspaces of symmetric or skewsymmetric vectors Fn± are invariant under B. We show that the restrictions of a symmetric T-Bezoutian to Fn± can be characterized by another kind of Bezoutians which is introduced next. Let p, q ∈ Fn+2 be either both symmetric or both skewsymmetric. Then Bsplit (t, s) =

p(t)q(s) − q(t)p(s) (t − s)(ts − 1)

is a polynomial in t and s. The n × n matrix with the generating polynomial Bsplit (t, s) will be called split Bezoutian of p(t) and q(t) and denoted by Bezsplit (p, q) . Obviously, Bezsplit (p, q) is a symmetric and centrosymmetric matrix. If p and q are symmetric, then we will speak about a split Bezoutian of (+)-type and if these vectors are skewsymmetric about a split Bezoutian of (−)-type. Instead of Bsplit we write B+ or B− , respectively.

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The columns and rows of a split Bezoutian of (+)-type are all symmmetric and of a split Bezoutian of (−)-type are all skewsymmetric, so that its rank is at most 12 (n + 1) in the (+) case and 12 n in the (−) case. As an example we consider the case p(t) = t2k + 1 ∈ F2k+1 (t) and q(t) = tk ∈ F2k+1 (t). In this case B+ (t, s) =

tk − sk (ts)k − 1 (t2k + 1)sk − tk (s2k + 1) = (t − s)(ts − 1) t−s ts − 1

= (tk−1 + tk−2 s + · · · + sk−1 )(1 + ts + · · · + tk−1 sk−1 ) . For k = 3, the matrix with this generating polynomial is ⎡ ⎢ 1 ⎢ 1 B+ = ⎢ ⎢ ⎣ 1

⎤

1 1 1 1

⎥ ⎥ 1 ⎥ ⎥. ⎦

1 For a general symmetric p = (pi )7i=1 ∈ F7 and q as before, q = e4 , the split Bezoutian of p and q is given by ⎡ ⎢ ⎢ B+ = ⎢ ⎢ p1 ⎣

p1 p2 p1

p1 p2 p 1 + p3 p2 p1

⎤ p1 p2 p1

⎥ ⎥ p1 ⎥ ⎥. ⎦

Moreover, from this special case it is clear how the split Bezoutian of a general and q = ek+1 looks like. p ∈ F2k+1 + Recall that P± = 12 (In ± Jn ) are the projections from Fn onto Fn± along Fn∓ . Our aim is to describe BP± = P± BP± for a symmetric T-Bezoutian B. As we know from Proposition 2.11, a symmetric T-Bezoutian B is the T-Bezoutian of a symmetric vector u+ ∈ Fn+1 and a skewsymmetric vector v− ∈ Fn+1 + − . From these vectors we form the polynomials p± (t) = (t ± 1)u+ (t) and q± (t) = (t ∓ 1)v− (t) . Clearly, p+ and q+ are symmetric, and p− and q− are skewsymmetric. Proposition 2.15. The symmetric T-Bezoutian B = BezT (u+ , v− ) can be represented as B = B+ + B− , where B± = BP± and 1 B± = ∓ Bezsplit (p± , q± ) . 2

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47

Proof. We compute the generating polynomial B+ (t, s) of B+ = BP+ . By deﬁnition we have 1 B+ (t, s) = (B(t, s) + B(t, s−1 )sn−1 ) 2

1 u+ (t)v− (s) + v− (t)u+ (s) u+ (t)v− (s) − v− (t)u+ (s) + = − 2 1 − ts t−s =

−

1 (t + 1)u+ (t)(s − 1)v− (s) − (t − 1)v− (t)(s + 1)u+ (s) . 2 (t − s)(ts − 1)

This is just the generating polynomial of the matrix − 21 Bezsplit (p+ , q+ ). The other case is proved analogously. 12. Relations between H- and T-Bezoutians. There is a simple relation between H- and T-Bezoutians, namely BezT (u, v) = −BezH (u, v)Jn . More general relations can be described with the help of Frobenius-Fischer transformations. Analogously to Theorem 2.7 we obtain the following. Theorem 2.16. For any ϕ ∈ GL(F2 ), the transformation Φ : B → Kn (ϕ)BKn (J2 ϕ)T maps T-Bezoutians into H-Bezoutians. Moreover, Kn (ϕ)BezT (u, v)Kn (J2 ϕ)T =

1 1 ) = ) , (2.18) u, v v, u BezH ( BezH ( det(ϕJ2 ) det ϕ

= Kn+1 (ϕ)u and v = Kn+1 (ϕ)v . where u In the case F = C it is of particular interest to describe congruence transformations that transform Hermitian T-Bezoutians into real symmetric H-Bezoutians, in other words to describe a coordinate transformation that transforms Hermitian T-Bezoutian forms into real quadratic H-Bezoutian forms. The transformation Φ has this property if and only if Kn (J2 ϕ)T = K n (ϕ)∗ (up to multiples of I2 ), which a c is equivalent to J2 ϕ = ϕ. Suppose that ϕ = , then this is equivalent to b d a = b and c = d. It can be easily checked that ϕ has this property if and only if at + b maps the unit circle onto the real line the linear fractional function ϕ(t) = ct + d (compare Section 1.5). Hence we have the following. Corollary 2.17. If ϕ(t) maps the unit circle onto the real line, then the transformation Φ : B → Kn (ϕ)BKn (ϕ)∗ maps Hermitian T-Bezoutians into real symmetric H-Bezoutians. In particular, the signatures of B and Φ(B) coincide.

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3. Resultant matrices and matrix representations of Bezoutians In this section we show that Bezoutians are closely related to resultant matrices and that the relations between these two classes can be used to derive important matrix representations of Bezoutians. We present two kinds of relations between resultant matrices and Bezoutians. The ﬁrst is due to Kravitsky and Russakovsky, the second an interpretation of Bezoutians as Schur complements in resultant matrices. The resultant matrix Res(u, v) of two polynomials u(t) ∈ Fm+1 , v(t) ∈ Fn+1 (t) was introduced in (2.4) as the (m + n) × (m + n) matrix

Mn (u)T . Res (u, v) = Mm (v)T In this section we restrict ourselves to the case m = n, which is no restriction of generality when speaking about nonsingularity, rank and related quantities. Recall that Res (u, v) is nonsingular if and only if the polynomials u(t) and v(t) are coprime and at least one of the leading coeﬃcients of u(t) or v(t) is not zero. 1. Kravitsky-Russakovsky formulas. To begin with we generalize the resultant concept. Let u(t) and v(t) be polynomials of degree n. The p-resultant matrix (p = 0, 1, . . . ) of u(t) and v(t) is, by deﬁnition, the (2n + 2p) × (2n + p) matrix

Mn+p (u)T Resp (u, v) = . Mn+p (v)T In the case p = 0 we have the resultant matrix in the former sense. For the sequel it is important to observe that

u(t)n+p (t) Resp (u, v)2n+p (t) = , (3.1) v(t)n+p (t) where m (t) = (ti−1 )m i=1 . Theorem 3.1. Let u(t) and v(t) be polynomials of degree n. Then 1.

Resp (u, v)T

O −Jn+p

Jn+p O

⎡

O Resp (u, v) = ⎣ O BH

O O O

⎤ −BH O ⎦ , O

(3.2)

where BH = BezH (u, v) , and 2.

Resp (u, v)T

In+p O

O −In+p

where BT = BezT (u, v) .

⎡

BT Resp (vJ , uJ ) = ⎣ O O

O O O

⎤ O O ⎦ , −BT

(3.3)

Introduction to Bezoutians

49

Proof. We compare the generating polynomials of the right-hand and of the lefthand sides. According to (3.1) we have

O Jn+p Resp (v, u) 2n+p (s) 2n+p (t)T Resp (u, v)T −Jn+p O (u(t)v(s) − v(t)u(s))n+p (t)T Jn+p n+p (s) tn+p − sn+p = (u(t)v(s) − v(t)u(s)) t−s = (tn+p − sn+p )BezH (u, v)(t, s) , =

which is the polynomial form of the ﬁrst assertion. To prove the second relation we observe that (3.1) implies

In+p O Resp (vJ , uJ )2n+p (s) 2n+p (t)T Resp (u, v)T O −In+p 1 − (ts)n+p (u(t)vJ (s) − v(t)uJ (s)) . 1 − ts This leads to the second assertion. =

2. Matrix representations of Bezoutians. The Kravitsky-Russakovsky formulas (3.2) and (3.3) provide an elegant way to obtain matrix representations of Bezoutians in terms of triangular Toeplitz matrices. These formulas are very important in connection with inversion of Toeplitz and Hankel matrices. They represent so-called “inversion formulas”. Note that from computational point of view the formulas presented here are not the most eﬃcient ones. Other, more eﬃcient formulas for the cases F = C or F = R can be found in [38], [40], [41], [42], [43]. We deﬁne, for u = (ui )n+1 i=1 , the lower triangular n × n Toeplitz matrix ⎡ ⎤ u1 ⎢ ⎥ .. T (u) = ⎣ ... ⎦. . un

...

u1

Note that T (u) is the T-Bezoutian B− (u) of u(t) and tn , which was introduced in (2.14). Note also that the matrix T (u) is related to the H-Bezoutian B(u) deﬁned by (2.2) and the T-Bezoutian B+ (u) deﬁned by (2.14) via B(u) = Jn T (uJ ) ,

B+ (u) = −T (uJ )T .

Furthermore, let us mention that we have commutativity T (u1 )T (u2 ) = T (u2 )T (u1 ) T

and the relation T (u) = T (u)J . The nonsingular matrices T (u) form a commutative subgroup of GL(Fn ) . With this notation the resultant matrix Res(u, v) for u, v ∈ Fn+1 can be written in the form

T (u)T T (uJ ) Res(u, v) = . T (v)T T (vJ )

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The application of Theorem 3.1 for p = 0 leads now to the following. Theorem 3.2. The H-Bezoutian of two polynomials u(t), v(t) ∈ Fn+1 admits 1. the representations BezH (u, v) = T (v)Jn T (uJ ) − T (u)Jn T (vJ ) and BezH (u, v) = T (uJ )T Jn T (v)T − T (vJ )T Jn T (u)T . 2. the representations BezT (u, v) = T (u)T (vJ )T − T (v)T (uJ )T and BezT (u, v) = T (vJ )T (u)T − T (uJ )T (v)T . 3. Bezoutians as Schur complements. We assume that the polynomial u(t) has degree n. Then the matrix T (uJ ) is nonsingular. Now the second expression for BezH (u, v) in Theorem 3.2,1 can be written in the form C = T (uJ )−1 Jn BezH (u, v) = T (v)T − T (vJ )T (uJ )−1 T (u)T . We see that C is the Schur complement of the left upper block in

O In T (uJ ) T (u)T R = Res (u, v) = . In O T (vJ ) T (v)T Recall that the concept of Schur complement is deﬁned in connection with the factorization of a block matrix

In A−1 B O In A O A B = , G= CA−1 In O In O D − CA−1 B C D where A is assumed to be invertible. Here D − CA−1 B is said to be the Schur complement of A in G. Applying this factorization to our case we obtain the following. Proposition 3.3. Let u(t) ∈ Fn+1 (t) be a polynomial of degree n , v(t) ∈ Fn+1 (t). Then the resultant of u(t) and v(t) can be represented in the form

T (uJ )−1 T (u)T In O O In T (uJ ) . Res (u, v) = T (vJ ) T (uJ )−1 Jn O BezH (u, v) In O From this proposition we see that Res(u, v) is nonsingular if and only if BezH (u, v) has this property. Hence BezH (u, v) is nonsingular if and only if the polynomials u(t) and v(t) are coprime. Taking (2.10) into account we conclude the following. Corollary 3.4. The nullity of BezH (u, v) is equal to the degree of the greatest common divisor of u(t) and v(t). Clearly, the same is also true for T-Bezoutians.

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51

4. Inverses of Hankel and Toeplitz matrices The most striking property of H- and T-Bezoutians is that inverses of Hankel and Toeplitz matrices belong to these classes. In view of Theorem 3.2 a consequence of this fact is that inverses of Toeplitz and Hankel matrices can be represented as product sum of triangular Toeplitz matrices, which is important for fast matrixvector multiplication. Later, in Section 7.6 and Section 8.7 we will see that, vice versa, inverses of H- and T-Bezoutians are Hankel or Toeplitz matrices, respectively. Let us start with the Hankel case. 1. Inverses of Hankel matrices. Let Hn = [ si+j−1 ]ni,j=1 be a nonsingular Hankel matrix. Besides Hn we consider the (n − 1) × (n + 1) Hankel matrix ∂Hn which is obtained from Hn after deleting the last row and adding another column on the right so that the Hankel structure is preserved. That means ⎤ ⎡ . . . sn+1 s1 ⎥ ⎢ .. . ∂Hn = ⎣ ... (4.1) ⎦. .. . sn−1

...

s2n−1

For Hn is nonsingular, ∂Hn has a two-dimensional nullspace. A basis { u, v } of the nullspace of ∂Hn will be called fundamental system for Hn . We consider for ﬁxed s ∈ F the linear system of equations Hn xs = n(s) ,

(4.2)

where n (s) is introduced in (1.1). It can be checked that

xs 0 = n−1 (s) and ∂Hn = sn−1 (s) . ∂Hn 0 xs

xs 0 −s belongs to the kernel of ∂Hn . In polynomial language, Hence 0 xs this means that there are constants as and bs such that (t − s)xs (t) = as u(t) − bs v(t) . Now we consider s as a variable. From (4.2) it is clear that xs (t) = n (t)T Hn−1 n (s) is a polynomial in s of degree n − 1. (It is just the generating polynomial of the matrix Hn−1 .) We conclude that as = a(s) and bs = b(s) ∈ Fn+1 (s). Thus, Hn−1 is a quasi-H-Bezoutian. According to Corollary 2.5, this implies that Hn−1 is an H-Bezoutian, which means that a(t) = γv(t) and b(t) = γu(t), and Hn−1 = γ BezH (u, v) for some nonzero constant γ. It remains to compute γ. For this we introduce the 2 × (n + 1) matrix

sn . . . s2n−1 s2n . F = 0 ... 0 1 Here s2n ∈ F is arbitrary. A fundamental system {u, v} will be called canonical if F [ u v ] = I2 .

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Let {u, v} be canonical. Then, in particular, u := eTn+1 u = 0 and v = eTn+1 v = 1. Furthermore, if we consider u as a vector in Fn , then it is just the last column of Hn−1 , i.e., Hn u = en . (4.3) We compare u with the last column of BezH (u, v), which is equal to vu−uv = −u −z , where (cf. Theorem 3.2). Thus, γ = −1. Note that v is of the form v = 1 z is the solution of the system Hn z = g

with

g = (sn+i )ni=1 .

(4.4)

Hereafter we need the following fact. Proposition 4.1. Let the equations (4.3) and (4.4) be solvable. Then Hn is nonsingular. Proof. Assume that Hn is singular, and let v = (vj )nj=1 be a nontrivial vector such that Hn v = 0 . Then applying vT from the left side to the equations (4.3) and (4.4) leads to vT Hn u = vT en = 0 and vT Hn z = vT g = 0, which means, in particular, that vn = 0 . Taking into account Hn Sn − SnT Hn = en gT − geTn we conclude (Sn v)T Hn = 0. Repeating the above arguments for the Sn v instead of v shows that vn−1 = 0, and so on. Finally we have v = 0 which is a contradiction. Thus, the nonsingularity of Hn is proved. Now we consider a general fundamental system {u, v}. The matrix ϕ = F [ u v ] is nonsingular. In fact, suppose it is singular. Then there is a nontrivial linear combination w(t) of u(t) and v(t) such that F w = 0. In particular the highest-order coeﬃcient vanishes, i.e., w ∈ Fn . Since w ∈ ker ∂Hn we conclude that Hn w = 0, which means that Hn is singular. The columns of [ u v ]ϕ−1 form now a canonical fundamental system. It remains to apply Lemma 2.2 to obtain the following. Theorem 4.2. Let {u, v} be a fundamental system for Hn . Then Hn−1 =

1 BezH (v, u) , det ϕ

(4.5)

where ϕ = F [ u v ]. Since BezH (u, v) is nonsingular, the polynomials u(t) and v(t) must be coprime (cf. Corollary 3.4). Hence the following is true. Corollary 4.3. If {u, v} is a fundamental system for a nonsingular Hankel matrix, then the polynomials u(t) and v(t) are coprime.

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53

2. Characterization of fundamental systems. There are several possibilities to characterize fundamental systems via solutions of special linear systems. We are mainly interested in characterizations by vectors that will be computed recursively using Levinson algorithms. In the Hankel case these vectors are the last columns xk of Hk−1 or alternatively the monic solutions uk of the Yule-Walker equations Hk uk = ρk ek , where ρk is so that eTk uk = 1 . It is convenient to consider an (n+1)×(n+1) extension Hn+1 = [ si+j−1 ]n+1 i,j=1 . The matrix Hn+1 is for almost all choices of s2n and s2n+1 nonsingular. In fact, Hn+1 is nonsingular if the Schur complement of the leading principal submatrix Hn in Hn+1 is nonsingular. This Schur complement is equal to s2n+1 − gT Hn−1 g. That means, for any s2n thereis only one value of s2n+1 for which Hn+1 is singular. un , which will be also denoted by un , and the vector Now, since the vector 0 un+1 are linearly independent and belong both to the kernel of ∂Hn they form a 1 fundamental system for Hn . To compute the factor in (4.5) we observe that det ϕ

ρn 0 . F un un+1 = 0 1 For the corresponding vectors xn and xn+1 we ﬁnd that

1 0 , F xn xn+1 = 0 ξn+1 where ξn+1 is the last component of xn+1 . Corollary 4.4. The inverse of the Hankel matrix Hn is given by 1 1 BezH (un+1 , un ) = BezH (xn+1 , xn ) . Hn−1 = ρn ξn+1 3. Christoﬀel-Darboux formula. We compare the ﬁrst Bezoutian formula for Hankel matrix inversion of Corollary 4.4 with the UL-factorization of Hn−1 (see, e.g., [33]) which can be written in polynomial language as n 1 Hn−1 (t, s) = uk (t)uk (s) . ρk k=1

We conclude n 1 un+1 (t)un (s) − un (t)un+1 (s) 1 uk (t)uk (s) = . ρk ρn t−s

(4.6)

k=1

This relation is called Christoﬀel-Darboux formula. It is important in the theory of orthogonal polynomials. 4. Inverses of Toeplitz matrices. The proof of the fact that inverses of Toeplitz matrices are T-Bezoutians follows the same lines as that for Hankel matrices. We

54

G. Heinig and K. Rost

n introduce the (n − 1) × (n + 1) Toeplitz matrix ∂Tn obtained from Tn = [ ai−j ]i,j=1 after deleting the ﬁrst row and adding another column to the right by preserving the Toeplitz structure, ⎤ ⎡ a0 . . . a2−n a1−n a1 ⎢ .. .. ⎥ . .. .. (4.7) ∂Tn = ⎣ ... . . . ⎦ .

an−1

an−2

...

a0

a−1

If Tn is nonsingular, then ∂Tn has a two-dimensional nullspace. Each basis of this subspace is called fundamental system for Tn . The role of the matrix F is taken by

a0 . . . a1−n a−n F = , 0 ... 0 1 where a−n is arbitrary. Theorem 4.5. Let {u, v} be a fundamental system for Tn . Then Tn−1 =

1 BezT (u, v) , det ϕ

where ϕ = F [ u v ]. The Toeplitz analogue of Proposition 4.1 is now as follows. Proposition 4.6. Let the equations T n y = e1

and

Tn z = f J

with f = (a−i )ni=1 be solvable. Then Tn is nonsingular. Taking into account that Tn Sn − Sn Tn = e1 f T − f J eTn the proof of this proposition is analogous to that one of Proposition 4.1. 5. Characterization of fundamental systems. In the Toeplitz case the Levinson + −1 algorithm computes recursively the ﬁrst and last columns x− or k and xk of Tk ± alternatively the solutions uk of the Yule-Walker equations − Tk u − k = ρk e1 ,

and

+ Tk u+ k = ρk ek ,

and

eTk u+ k = 1.

(4.8)

where ρ± k ∈ F are so that eT1 u− k = 1

− (In other words u+ k (t) is assumed to be monic and uk (t) comonic, which means − J that (uk ) (t) is monic.) So it is reasonable to describe the fundamental system with these vectors.

− 0 xn belong to the nullspace of and It can easily be seen that x+ 0 n ∂Tn and in the case where Tn−1 is nonsingular they are linearly independent.

Introduction to Bezoutians

Thus, they form a fundamental system. Likewise fundamental system. We ﬁnd that

−

un 0 ρn ∗ F , = 0 u+ 0 1 n

F

x− n 0

55 u− n 0

0 x+ n

and

=

1 0

0 u+ n

∗ ξn

form a

,

+ where ξn is the ﬁrst component of x− n which equals the last component of xn . + − Consequently, ρn = ρn = ρn . We will have a problem with these systems, if the submatrix Tn−1 is singular. For example, in this case the solution u+ n does not exist and ξn = 0. For this reason we also consider, like in the Hankel case, an (n + 1) × (n + 1) Toeplitz extension Tn+1 = [ ai−j ]n+1 i,j=1 . This extension is nonsingular for almost all choices of a±n . The proof of this fact is, however, less trivial than in the Hankel case. The Schur complement of Tn in Tn+1 is given by

σ = a0 − (g+ + an e1 )T Tn−1 (g− + a−n e1 ) , where g± = [ 0 a±(n−1) . . . a±1 ]T . Hence −σ = ξan a−n + η− an + η+ a−n + ζ ,

(4.9)

T −1 where ξ = eT1 Tn−1e1 = eT1 x− n , ζ = g+ Tn g− − a0 , and T + xn , η− = eT1 Tn−1 g− = g−

T −1 T − η+ = g + Tn e1 = g+ xn .

If ξ = 0, which is equivalent to the nonsingularity of Tn−1 , the set of pairs (an , a−n ) for which Tn+1 is singular is a quadratic curve in F2 . (Choosing, for example, an = a−n there are at most 2 values of an for which Tn+1 is singular.) We show that if ξ = 0, then η± = 0. In fact, in the case where ξ = 0 we have Tn SnT x− n = η+ en . Since Tn is assumed to be nonsingular, we have η+ = 0. Analogously, η− = 0. That means in the case ξ = 0 the pairs (an , a−n ) for which Tn+1 is singular are on the graph of a polynomial of ﬁrst degree. (Choose, for example, a−n = 0 , then Tn+1 is nonsingular with the exception of one value of an .) Let now Tn+1 be a nonsingular Toeplitz extension of Tn , x− n+1 the ﬁrst and + −1 . Furthermore, let u± be the solutions of the corxn+1 the last column of Tn+1 n+1 + , x responding Yule-Walker equations(4.8) for k = n + 1. Then {x− n+1 n+1 } and − + {un+1 , un+1 } are fundamental systems for Tn and

− − 1 0 ρn+1 0 + + F xn+1 xn+1 = . , F un+1 un+1 = ∗ 1 ∗ ξn+1 (4.10) Corollary 4.7. The inverse of the Toeplitz matrix Tn is given by Tn−1 =

1 ξn+1

+ BezT (x− n+1 , xn+1 ) =

1 ρn+1

+ BezT (u− n+1 , un+1 ) .

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6. Inverses of symmetric Toeplitz matrices. We discuss now the case of a symmetric Toeplitz matrix Tn . Let Tn+1 be a symmetric Toeplitz extension of Tn . Since in this case g+ = g− we have in (4.9) η+ = η− . From this we conclude that Tn+1 is nonsingular with the exception of at most two values of an . Thus, we may assume that Tn+1 is nonsingular. − + J J Since we have xn+1 := x+ n+1 = (xn+1 ) , the vectors wn+1 = xn+1 + xn+1 − J and wn+1 = xn+1 − xn+1 form a fundamental system consisting of a symmetric ± ± are the solutions of Tn+1 wn+1 = and a skewsymmetric vector. The vectors wn+1 en+1 ± e1 and

− −1 1 + F wn+1 wn+1 = . − + (0) wn+1 (0) −wn+1 Corollary 4.8. The inverse of a nonsingular symmetric Toeplitz matrix Tn is given by 1 − + Tn−1 = BezT (wn+1 , wn+1 ), γ − + (0) − wn+1 (0). where γ = wn+1 One can show that for solving a system Tn z = b it is suﬃcient to compute − the vectors wk+ . So it is reasonable to ask whether it is possible to describe wn+1 + in terms of wk . The following proposition gives an answer to this question. Let Tn+2 be a nonsingular (n + 2) × (n + 2) symmetric Toeplitz extension of Tn+1 and ± ± wn+2 the solutions of Tn+2 wn+2 = en+2 ± e1 . ± Proposition 4.9. The polynomials wn+1 are given by + twn+ (t) − c± wn+2 (t) , (4.11) 1±t + + + where wn+2 (1) = 0 and c− = wn+ (1)/wn+2 (1). If n is odd, then wn+2 (−1) = 0 + + and c+ = −wn+ (−1)/wn+2 (−1). If n is even, then wn+2 (−1) = 0 and c+ is not + alone. determined by wn+ and wn+2 ± wn+1 (t) =

Proof. We have

Tn+2

± wn+1 0

⎡ 0

± wn+1

⎢ ⎢ =⎢ ⎢ ⎣

±1 0 0 1 a±

±a± ±1 0 0 1

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

⎤

⎡

0 Tn+2 ⎣ wn+ 0

+ wn+2

⎡

⎢ ⎢ ⎦=⎢ ⎢ ⎣

b 1 0 1 b

1 0 0 0 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

for some a± , b ∈ F. Consequently, ⎤ ⎡

±

0 0 wn+1 + ± = ⎣ wn+ ⎦ − c± wn+2 ± wn+1 0 0 ± for some c± ∈ F . Writing this in polynomial language, we see that wn+1 (t) ± ± + + twn+1 (t) = twn (t) − c± wn+2 (t) and obtain (4.11).

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57

+ (t) To prove the rest of the proposition we recall that the polynomials wn+1 − and wn+1 (t) form a fundamental system. Therefore, they are coprime. Suppose + + − that wn+2 (1) = 0. Then (4.11) implies wn+1 (1) = 0. But we have also wn+1 (1) = − + 0, since wn+1 is skewsymmetric. This contradicts the coprimeness of wn+1 (t) and − + + (t). Consequently, wn+2 (1) = 0. Analogously, if n is odd and wn+2 (−1) = 0, wn+1 − + + then (4.11) implies wn+1 (−1) = 0. But we have also wn+1 (−1) = 0, since wn+1 is + symmetric and has an even length. This contradiction shows that wn+2 (−1) = 0. If n is even, then Tn is not completely determined by its restriction to symmetric + + vectors. That means wn+1 is not completely given by wn+ and wn+2 .

If n is even, then the constant c+ can be obtained by applying a test functional, which could be the multiplication by any row of Tn+1 . 7. Inverses of skewsymmetric Toeplitz matrices. In the case of a nonsingular skewsymmetric Toeplitz matrix Tn, n = 2m , the Levinson-type algorithm can be used to compute vectors spanning the nullspace of T2k−1 for k = 1, . . . , m. So it is reasonable to ask for a fundamental system {u, v} consisting of vectors of this kind. J Let x be any vector spanning the nullspace of Tn−1 . From the relation Tn−1 = J −Tn−1 follows that also the vector x belongs to the nullspace of Tn−1 . Thus x is either symmetric or skewsymmetric. We show that the latter is not possible. Lemma 4.10. The vector x is symmetric. Proof. Let fj denote the jth row of Tn−1 , n = 2m . State that the row fm in the middle of Tn−1 is skewsymmetric. We introduce vectors fj± = fj ∓ fn−j for j = , 1, . . . , m − 1 . Then the fj+ are symmetric, the fj− are skewsymmetric, fj± ∈ Fn−1 ± and the system Tn−1 v = O is equivalent to fj± v = 0 for j = 1, . . . , m − 1 and = n2 , there exists a symmetric vector v = O such that fm v = 0 . Since dim Fn−1 + + fj v = 0 for j = 1, . . . , m − 1 . Since, obviously, fj− v = O and fm v = 0 we have Tn−1 v = O . Taking into account that dim ker Tn−1 = 1 we conclude that x = cv for some c ∈ F . Thus, x ∈ Fn−1 . + Now, by Lemma 4.10, x is symmetric and ⎤ ⎡ 0 u = ⎣ x ⎦ ∈ ker ∂Tn . 0 (Since we do not want to assume that Tn−2 is nonsingular, we cannot assume that x is monic.) Furthermore, let Tn+1 be any (n+1)×(n+1) skewsymmetric Toeplitz extension of Tn and v a (symmetric) vector spanning the nullspace of Tn+1 . Since Tn is nonsingular, we may assume that v is monic. Now {u, v} is a fundamental system, and

γ 0 , γ = [ a1 . . . an−1 ]x . F u v = 0 1

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G. Heinig and K. Rost

(Here γ = 0 since otherwise

0 x

belongs to the kernel of Tn .) Thus we obtain

the following. Corollary 4.11. The inverse of the nonsingular skewsymmetric Toeplitz matrix Tn is given by 1 Tn−1 = BezT (u, v) . γ 8. Inverses of Hermitian Toeplitz matrices. Finally we discuss the case of a nonsingular Hermitian Toeplitz matrix Tn . Besides Tn we consider an (n + 1) × (n + 1) Hermitian Toeplitz extension Tn+1 of Tn . With similar arguments as above one can show that for almost all values of an the matrix Tn+1 is nonsingular, so we may assume this. In the Hermitian case we have for the ﬁrst and last columns + −1 x− n+1 , xn+1 of Tn+1 that + # xn+1 := x− n+1 = (xn+1 )

and for the solutions u± n+1 of the Yule-Walker equations + # un+1 := u− n+1 = (un+1 ) .

Taking Corollary 4.7 into account we obtain 1 1 BezT (xn+1 , x# BezT (un+1 , u# (4.12) Tn−1 = n+1 ) = n+1 ) , ξn+1 ρn+1 where ξn+1 is the ﬁrst component of xn+1 and ρn+1 is so that un+1 (t) is comonic. In the Levinson-type algorithm described in [53], [47] not the vectors xk are computed but the solutions of the equations Tk qk = e, where e is the vector all components of which are equal to 1. For an inversion formula we need the vectors qn and qn+1 . Since qn+1 and qn are conjugate-symmetric, qn+1 (t) − tqn (t) is not identically equal to zero. Hence xn+1 (t) = b(qn+1 (t) − tqn (t))

(4.13)

for some nonzero b ∈ C. Besides qn we consider the coeﬃcient vector w of w(t) = i (t− 1)qn (t), which is obviously conjugate-symmetric. Proposition 4.12. The inverse of a nonsingular Hermitian Toeplitz matrix Tn is given by i 1 (4.14) Tn−1 = BezT (w, qn+1 ) − qn qTn , c c where c is the real constant qn+1 (1) − qn (1). Proof. We insert (4.13) into (4.12) and obtain, after an elementary calculation, ξn+1 formula (4.14) with c = = 0. Taking into account that qn (t) = (Tn−1 e)(t) = |b|2 1 Tn−1 (t, 1) and that, due to (4.14), Tn−1(t, 1) = qn (t)(qn+1 (1) − qn (1)) we ﬁnd c that c = c = qn+1 (1) − qn (1).

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9. Solution of systems. The formulas for the inverses of Toeplitz and Hankel matrices presented in this section can be used in combination with the matrix representations of Bezoutians to solve Toeplitz and Hankel systems. This is in particular convenient if systems have to be solved with diﬀerent right-hand sides and one and the same coeﬃcient matrix. The advantage compared with factorization methods is that only O(n) parameters have to be stored. The application of the formulas requires 4 matrix-vector multiplications by triangular Toeplitz matrices. If these multiplications are carried out in the classical way, then 2 n2 multiplications and 2 n2 additions are needed, which is more than, for example, if back substitution in the LU-factorization is applied. However, due to the Toeplitz structure of the matrices there are faster methods, actually methods with a complexity less than O(n2 ), to do this. In the cases F = C and F = R the Fast Fourier and related real trigonometric transformations with a computational complexity of O(n log n) can be applied.

5. Generalized triangular factorizations of Bezoutians In this section we describe algorithms that lead to a generalized UL-factorization of Bezoutians. In the case of H-Bezoutians the algorithm is just the Euclidian algorithm. n+1 1. Division with remainder. Suppose that u = (ui )n+1 ,v = (vi )m+1 i=1 ∈ F i=1 ∈ m+1 F , m ≤ n , and that the last components of u and v are not zero. Division with remainder means to ﬁnd polynomials q(t) ∈ Fn−m+1 (t) and r(t) ∈ Fm (t) such u(t) = q(t)v(t) + r(t) . (5.1)

In matrix language this means that we ﬁrst solve the (n − m + 1) × (n − m + 1) triangular Toeplitz system ⎡ ⎤ ⎡ ⎤ vm+1 . . . vn−2m+1 um+1 ⎢ ⎥ ⎢ ⎥ .. .. .. ⎣ ⎦q = ⎣ ⎦, . . . vm+1

un+1

/ {1, . . . , m + 1}. With the notation (2.3) we ﬁnd r via where we put vi = 0 for i ∈

r = u − Mn−m+1 (v)q . 0 2. Factorization step for H-Bezoutians. We clarify what means division with remainder in terms of the H-Bezoutian. From (5.1) we obtain for B = BezH (u, v) q(t) − q(s) r(t)v(s) − v(t)r(s) v(s) + , t−s t−s which can be written in the form B(t, s) = v(t)

BezH (u, v) = Mn−m (v)B(q)Mn−m (v)T + BezH (r, v) ,

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where B(q) is deﬁned in (2.2). This is equivalent to

Im O Im BezH (r, v) Mn−m (v) BezH (u, v) = O B(q) O O

T Mn−m (v)

.

(5.2) Note that in this equation the left factor is upper triangular and the right factor is lower triangular. 3. Euclidian algorithm. The Euclidian algorithm is the successive application of division with remainder. We set u0 (t) = u(t) and u1 (t) = v(t). Then, for i = 1, 2, . . . we ﬁnd −ui+1 (t) as remainder of the division of ui−1 (t) by ui (t), i.e., ui+1 (t) = qi (t)ui (t) − ui−1 (t) .

(5.3)

We take the minus sign with ui+1 (t) for convenience. With this deﬁnition one factorization step (5.2) reduces BezH (ui−1 , ui ) to BezH (ui , ui+1 ), without the minus to BezH (ui+1 , ui ). If for some i = l we have ul+1 (t) = 0, then the algorithm is terminated and ul (t) is the greatest common divisor of u(t) and v(t). There are modiﬁcations of the Euclidian algorithm with diﬀerent normalizations. One could assume, for example, that the polynomials ui (t) are monic or that the polynomials qi (t) are monic. For these two possibilities one has to admit a constant factor at ui−1 . 4. Generalized UL-factorization of H-Bezoutians. Applying (5.2) successively to the polynomials ui (t) appearing in the Euclidian algorithm (5.3) we arrive at the following. Theorem 5.1. Let u(t), v(t) ∈ Fn+1 (t) where u(t) has a nonzero leading coeﬃcient. Then BezH (u, v) admits the factorization BezH (u, v) = U DU T

(5.4)

Id Mnl (ul ) . . . Mn1 −n2 (u2 ) U= Mn−n1 (u1 ) , O O O ni = deg ui (t), d is the degree of the greatest common divisor of u(t) and v(t), and (5.5) D = diag ( Od , B(ql ), . . . , B(q1 ) ) . where

Note that U is a nonsingular upper triangular matrix, B(qi ) are nonsingular upper triangular Hankel matrices deﬁned by (2.2). From the theorem we can again conclude that the nullity of BezH (u, v) is equal to d (cf. Corollary 3.4). Let us discuss the case, in which u(t) and v(t) are coprime and all polynomials qi (t) have degree 1. This is called the generic case. Suppose that qi (t) = ρi t + δi . Then (5.4) turns into an UL-factorization of the H-Bezoutian in which, U is the

upper triangular matrix the kth row of which is equal to the transpose of un+1−k and D = diag (ρn+1−i )ni=1 . The UL-factorization of the H-Bezoutian 0k−1 exists if and only if the matrix BezH (u, v)J is strongly nonsingular. That means

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61

in the generic case the matrix has this property. The converse is also true, since in the non-generic case the matrix BezH (u, v)J has singular leading principal submatrices. 5. Inertia computation. It is an important consequence of Theorem 5.1 that the signature of a real H-Bezoutian can be computed via running the Euclidian algorithm. In fact, in the case of real polynomials u(t) and v(t) the matrix BezH (u, v) is congruent to the block diagonal matrix D given by (5.5). It remains to compute the signature of B(qi ). Let ρi denote the leading coeﬃcient of qi (t). Then the signature of B(qi ) is equal to the signature of ρi Jmi , mi = ni−1 − ni . This can be shown using a homotopy argument. Let H(t) = tρi Jmi + (1 − t)B(qi ) for 0 ≤ t ≤ 1. Then H(0) = B(qi ) and H(1) = ρi Jmi . Furthermore, H(t) is nonsingular for all t and depends continuously on t. Hence sgn H(t) is constant for 0 ≤ t ≤ 1. The signature of ρi Jmi is obviously equal to zero if mi is even and is equal to the sign of ρi if mi is odd. Applying Sylvester’s inertia law we conclude the following. Corollary 5.2. The signature of the real H-Bezoutian BezH (u, v) is given by sgn ρi , sgn BezH (u, v) = ni−1 −ni odd

where ρi are the antidiagonal entries of B(qi ). Since the Euclidian algorithm computes besides the signature s also the rank r of the H-Bezoutian, it gives a complete picture about the inertia of BezH (u, v), In BezH (u, v) = (s+ , s− , d), where s± =

n−d±s 2

and d = n − r .

6. Factorization step for T-Bezoutians in the generic case. We consider the problem of triangular factorization of a T-Bezoutian B = BezT (u, v), where u, v ∈ Fn+1 . This problem is more complicated than for H-Bezoutians, unless the matrix is strongly nonsingular. We introduce the 2 × 2 matrix

eT1 u eT1 v . Γ = Γ(u, v) = eTn+1 u eTn+1 v The case of nonsingular Γ is referred to as generic case, the case of singular Γ as non-generic case. In this subsection we consider the generic case. Observe that γ := B(0, 0) = det Γ. That means that γ is the entry in the left upper corner of B. Thus, we have the generic case if B is strongly nonsingular. Note that γ is also the entry in the right of B, due to the persymmetry of B. In the generic lower corner v = u v Γ−1 is of the form case, u

u1 0 v = u , u1 , v1 ∈ Fn . 0 v1

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According to Lemma 2.9 we have ) = u, v BezT (

1 BezT (u, v) . γ

= BezT ( ) we obtain Furthermore, for B u, v s) B(t,

u1 (t)v1 (s−1 )sn−1 − tsv1 (t)u1 (s−1 )sn−1 1 − ts = B1 (t, s) + v1 (t)u1 (s−1 )sn−1 , =

where B1 = BezT (u1 , v1 ). We also have s) = ts B1 (t, s) + u1 (t)v1 (s−1 )sn−1 . B(t, In matrix language this can be written as

T In−1 In−1 γ BezT (u1 , v1 ) 0 J v1 u1 BezT (u, v) = 0T 0T γ 0T or

T

In−1 In−1 γ 0T J v BezT (u, v) = u1 . (5.6) 1 0T 0 γ BezT (u1 , v1 ) 0T 7. LU-factorization of T-Bezoutians. Let B = BezT (u, v) be strongly nonsingular, which is equivalent to the strongly nonsingularity of B J , due to persymmetry. We can apply now the factorization step of the previous subsection, since the property of strongly nonsingularity is inherited after a factorization step. If we carry out the factorization step successively, then we obtain the following algorithm. We set u1 = u and v1 = v and ﬁnd recursively polynomials uk (t) and vk (t) via

1 0 uk+1 (t) vk+1 (t) = uk (t) vk (t) Γ−1 , (5.7) k 0 t−1 where Γk = Γ(uk , vk ). This algorithm has the same structure as the Schur algorithm for Toeplitz matrices. We call it also Schur algorithm. Like for Toeplitz −1 matrices, it can be slightly modiﬁed by replacing the matrix Γk by a matrix of 1 ∗ . This will reduce the number of operations. the form ∗ 1 To simplify the notation we agree upon the following. For a sequence (wj )nj=1 with wj ∈ Fn+1−j , by L(wj )nj=1 will be denoted the lower triangular matrix the kth column of which is equal to

0k−1 L(wj )nj=1 ek = . wk Now we conclude the following from (5.6). Theorem 5.3. Let B = BezT (u, v) be strongly nonsingular, and let uk (t) and vk (t) be the polynomials obtained by the Schur algorithm (5.7). Then B admits an LUfactorization B = LDU ,

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63

where L = L(ui )ni=1 ,

U = (L(vi )ni=1 )T

and D = diag ( γi−1 )ni=1 ,

γ i =

i

γj ,

γj = det Γj .

j=1

8. Non-generic case for T-Bezoutians. Now we consider the case where the matrix Γ = Γ(u, v) is singular. If Γ has a zero row, then u(t) and v(t) or uJ (t) and vJ (t) have a common factor t. Suppose that u(t) = tμ− u0 (t), v(t) = tμ− v0 (t), uJ (t) = tμ+ uJ0 (t), and vJ (t) = tμ+ v0J (t) such that Γ(u0 , v0 ) has no zero row. Then B(t, s) = tμ− sμ+ B0 (t, s), where B0 = BezT (u0 , v0 ) or, in matrix language ⎤ ⎡ O O O B = ⎣ O B0 O ⎦ , O O O where the zero matrix in the left upper corner is μ− × μ+ . Now we assume that Γ is singular but has no zero row. Then there is a 2 × 2 matrix Φ with det Φ = 1 such that the last column of ΓΦ is zero, but the ﬁrst column consists of nonzero elements. We set (t) v (t) = u(t) v(t) Φ. u ) = BezT (u, v). Furtheru, v Then, according to Lemma 2.9, we have B = BezT ( in the form more, let us write v ⎡ ⎤ 0ν− =⎣ w ⎦ v 0ν+ m+1 , m + ν+ + ν− = n, with nonzero ﬁrst with some vector w = (wi )m+1 i=1 ∈ F and last components. We apply now a two-sided division with remainder to ﬁnd polynomials q− (t) ∈ Fν− (t) , q+ (t) ∈ Fν+ +1 (t) , and r(t) ∈ Fm (t) such that

(t) = (tν− q+ (t) + q− (t))w(t) + tν− r(t) . u The vectors q± can be found by solving the triangular Toeplitz systems ⎤ ⎤ ⎡ ⎡ w1 u 1 ⎥ ⎢ .. ⎢ . ⎥ .. ⎦ q− = ⎣ .. ⎦ , ⎣ . . wν− . . . w1 u ν− ⎡ ⎡ ⎤ ⎤ u n−ν+ +1 wm+1 . . . wm−ν+ +1 ⎢ ⎢ ⎥ ⎥ .. .. .. ⎣ ⎦ q+ = ⎣ ⎦. . . . wm+1

u n+1

Then we have uJ (t) = (tν+ +1 qJ− (t) + qJ+ (t))wJ (t) + tν+ +1 rJ (t)

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and

B(t, s) =

=

w(t)

q+ (t)sν+ − qJ+ (s) q− (t) − tν− qJ− (s)s ν+ s + tν− 1 − ts 1 − ts

wJ (s)

r(t)wJ (s) − w(t)rJ (s)s ν+ s +tν− 1 − ts w(t) BezT (q− , eν− +1 )(t, s)sν+ + tν− BezT (q+ , e1 )(t, s) wJ (s) +tν− BezT (r, w)(t, s)sν+ .

In matrix form this can be written as ⎡

O O B− (q− ) B = Mν+ +ν− (w) Mν+ +ν− (wJ )T + ⎣ O B+ (q+ ) O O

O B1 O

⎤ O O ⎦, O

where B1 = BezT (r, w) is of order m, B+ (q+ ) = BezT (q+ , e1 ) and B− (q− ) = BezT (q, eν− +1 ) are of order ν± (cf. (2.14)), and the zero matrix in the left upper corner of the last term has size ν− × ν+ . 9. Hermitian T-Bezoutians. We discuss now the speciﬁcs of the case of an Hermitian T-Bezoutian B. Our main attention is dedicated to the question how to compute the signature, because this is the most important application of the procedure. First we remember that there are two possibilities to represent Hermitian T-Bezoutian. The ﬁrst is B = BezT (u, u# ) for a general vector u ∈ Cn+1 , the second is B = i BezT (u+ , u− ) for two conjugate-symmetric vectors u± (see Section 2.10). In the generic case we use the ﬁrst representation. In the representation (5.6) we have v1J = u1 , and γ is real. Thus, for a strongly nonsingular B, Theorem 5.3 provides a factorization B = LDL∗ . Consequently, sgn B =

n

sgn γi .

i=1

That means that the signature of B can be computed via the Schur algorithm in O(n2 ) operations. In the non-generic case, i.e., in the case where Γ is singular, we switch from the ﬁrst representation of B to the second one. This is done as follows. Suppose B is given as B = BezT (u, u# ). Then Γ is centro-Hermitian. Hence the homogeneous equation

0 α = Γ 0 β has a nontrivial conjugate-symmetric solution, which is a solution with β = α. We can assume that |α| = 1. We set

αi α Φ= −αi α

Introduction to Bezoutians and

u+ (t)

u− (t)

=

u(t) u# (t))

65

Φ.

Then u± are conjugate-symmetric and according to Lemma 2.9 we obtain B=

1 BezT (u+ , u− ) . 2i

= u+ and We can now apply the reduction step described in Section 5.8 for u = u− . Due to the Hermitian symmetry we have ν− = ν+ =: ν . The vector q# v − is just the vector q+ after cancelling its ﬁrst component. The vectors w and r , both considered as elements of Fm+1 , are conjugate-symmetric. This leads to ⎡ ⎤

O O O O B− (q− ) 2i B = M2ν (w) M2ν (w)∗ + ⎣ O B1 O ⎦ , (B− (q− ))∗ O O O O where B1 = BezT (r, w) is m × m. Taking into account that B− (q) and the zero matrices in the corners of the last term are ν × ν matrices, the sum of the ranks of the two terms on the right-hand side is equal to the rank of B. This is also true for the signature. The signature of the ﬁrst term is equal to zero. Hence sgn B = sgn B1 . If BezT (r, w) is singular, then we carry out another non-generic step. If BezT (r, w) is nonsingular we go over to the ﬁrst Bezoutian representation by introducing v = 12 (r − iw) and obtain B1 = BezT (v, v# ). Now we can apply a generic step. Summing up, we have described a procedure that computes the signature of an arbitrary Hermitian T-Bezoutian in O(n2 ) operations.

6. Bezoutians and companion matrices In this section we show that Bezoutians are related to functions of companion matrices. 1. Factorization of the companion. The companion matrix of the monic polynomial k−1 u t ∈ Fn+1 is, by deﬁnition, the n × n matrix u(t) = n+1 k=1 k ⎤ ⎡ 0 1 ⎥ ⎢ .. ⎥ ⎢ . C(u) = ⎢ ⎥ ⎣ 1 ⎦ −u1 −u2 . . . −un It is easy to show that the characteristic polynomial of C(u) , det(tIn − C(u)) , is equal to u(t). This is also a consequence of the following useful relation.

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Lemma 6.1. We have ⎡ 0 −1 ⎢ .. ⎢ . tIn − C(u) = ⎢ ⎣ 1 u1 (t) . . .

⎤ ⎥

⎥ u(t) ⎥ ⎦

−1 un−1 (t)

⎤

⎡ 1 ⎢ −t ⎢ ⎢ ⎣

In−1

1 .. .

⎥ ⎥ ⎥, ⎦

..

. −t

1 (6.1)

where uk (t) = uk+1 + uk+2 t + · · · + un+1tn−k . Proof. It is immediately ⎡ 1 ⎢ t ⎢ 2 ⎢ (tIn − C(u)) ⎢ t ⎢ .. ⎣ . tn−1

checked that 1 t .. .

1 .. .

..

tn−2

...

t

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎦ ⎣

. 1

⎤

−1

0

..

u(t)

u1 (t)

.

...

−1 un−1 (t)

⎥ ⎥ ⎥ ⎥. ⎥ ⎦

This equality can be rearranged to (6.1).

The polynomials uk (t) appearing in the ﬁrst factor of the right side of (6.1) are the Horner polynomials of uJ (t) . They satisfy the recursion uk (t) = tuk+1 (t) + uk+1 and can be represented as [ u1 (t) . . . un (t) ] = n (t)T B(u) . where B(u) is introduced in (2.2). For t0 ∈ F, the matrix t0 In − C(u) is a special case of a resultant matrix (cf. (2.4)). In fact, t0 In − C(u) = Res(t0 − t, u(t) + tn−1 (t0 − t)) . Since the resultant matrix is nonsingular if and only if the polynomials are coprime, we conclude again that t0 In − C(u) is singular if and only if t − t0 is a divisor of u(t), i.e., u(t0 ) = 0 . 2. Functional calculus. Before we continue with companions and Bezoutians we recall some general deﬁnitions and facts concerning functions of a matrix. Let A m uk tk−1 a polynomial. Then u(A) denotes the be an n × n matrix and u(t) = k=1

matrix u(A) =

m

uk Ak−1

k=1

in which we set A0 = In . The matrices of the form u(A) form a commutative matrix algebra and the transformation u(t) → u(A) is a linear operator and a ring homomorphism. In particular, if u(t) = u1 (t)u2 (t), then u(A) = u1 (A)u2 (A). If u(t) is the characteristic polynomial of A, then, according to the Cayley-Hamilton

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67

theorem, u(A) = O. Let a polynomial v(t) and the characteristic polynomial u(t) of A be coprime. Then the Bezout equation v(t)x(t) + u(t)y(t) = 1

(6.2)

has a solution (x(t), y(t)). Replacing t by A we obtain that v(A)x(A) = In . That means v(A) is nonsingular and x(A) is its inverse. 3. Barnett’s formula. The following remarkable formula is due to S. Barnett. Theorem 6.2. Let u(t), v(t) ∈ Fn+1 (t) and u(t) be monic. Then BezH (u, v) = B(u)v(C(u)) ,

(6.3)

where B(u) is introduced in (2.2). Proof. Due to linearity, it is suﬃcient to prove the formula for v(t) = ek (t) = tk−1 . We set Bk = BezH (u, ek ). Since Bk = B(u)C(u)k−1 is true for k = 1 we still have to show that Bk+1 = Bk C(u). Taking into account that C(u)n (s) = sn (s) − u(s)en , we obtain n (t)T Bk C(u)n (s) = sBk (t, s) − u(s)n (t)T Bk en = sBk (t, s) − tk−1 u(s) . (6.4) On the other hand, Bk+1 (t, s) =

(u(t)sk−1 − tk−1 u(s))s − tk−1 u(s) = sBk (t, s) − tk−1 u(s) . t−s

Comparing (6.4) and (6.5) we obtain the recursion Bk+1 = Bk C(u).

(6.5)

From this theorem we can conclude again (cf. Corollary 3.4) that the HBezoutian of u(t) and v(t) is nonsingular if u(t) and v(t) (u(t) ∈ Fn+1 (t) monic) are coprime and that its inverse is given by BezH (u, v)−1 = x(C(u))B(u)−1 ,

(6.6)

where x(t) is from the solution of the Bezout equation (6.2). In the next subsection we will show that x(C(u)) is actually a Hankel matrix. 4. Barnett’s formula for T-Bezoutians. Now we consider T-Bezoutians. Let u(t) be a comonic polynomial of degree ≤ n and Bk = BezT (u, ek ). Then Bk (t, s) = Bk+1 (t, s)s − tk−1 uJ (s) . This can be written as Bk = Bk+1 C(uJ ) . With the notation of (2.14) we obtain the Toeplitz analogue of Barnett’s formula. Theorem 6.3. Let u(t), v(t) ∈ Fn+1 (t), where u(t) is comonic. Then BezT (u, v) = B− (u)vJ (C(uJ )) .

68

G. Heinig and K. Rost In particular, for v(t) = 1 we obtain the equality B+ (u) = B− (u)C(uJ )n ,

(6.7)

which yields an LU-factorization of C(uJ )n and will be applied below to prove an inversion formula for T-Bezoutions.

7. Hankel matrices generated by rational functions In this section we consider Hankel matrices generated by rational functions and show that they are closely related to H-Bezoutians. We understand “rational functions” in an abstract sense as elements of the quotient ﬁeld of the ring of polynomials. But occasionally, in particular if we restrict ourselves to the case F = C, we interpret them in the analytic sense as functions deﬁned in F. By a proper rational function we mean a rational function for which the degree of the numerator polynomial is not greater than the degree of the denominator polynomial. We say that the representation f (t) = p(t) is in reduced form if u(t) u(t) and p(t) are coprime and u(t) is monic. This representation is unique. The degree of a proper rational function is the degree of the denominator polynomial in the reduced representation. 1. Generating functions of Hankel matrices. A proper rational function f (t) can be represented as f (t) = h0 + h1 t−1 + h2 t−2 + · · · .

(7.1)

If F = C, then (7.1) can be interpreted as the Laurent series expansion of f (t) at inﬁnity converging outside a disk with center 0. For a general ﬁeld F (7.1) has a meaning as quotient of two formal power series. The coeﬃcients hi can be obtained recursively by an obvious formula. For f (t) having a Laurent expansion (7.1), we set f (∞) = h0 and write f (t) = O(t−m ) if h0 = · · · = hm−1 = 0. Note that if f (t) is given by (7.1), then we have ∞ ∞ f (t) − f (s) t−k − s−k = =− hk hi+j−1 t−i s−j . t−s t−s i,j=1

(7.2)

k=1

This relation suggests the following deﬁnition. For n = 1, 2, . . . , the n × n Hankel matrix generated by f (t) is, by deﬁnition, the matrix Hn (f ) = [ hi+j−1 ]ni,j=1 . Let us point out that the entry h0 does not enter the deﬁnition of Hn (f ). If for some n × n Hankel matrix Hn there is a function f so that Hn = Hn (f ), then f (t) will be called generating function of Hn .

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69

Example 7.1. As an example, let us compute the Hankel matrices generated by a 1 (c ∈ F, m = 1, 2, . . . , 2n − 1). We denote partial fraction (t − c)m

1 1 Ln (c, m) = Hn (m − 1)! (t − c)m and write Ln (c) instead of Ln (c, 1). In view of 1 = t−1 + c t−2 + c2 t−3 + · · · t−c

(7.3)

Ln (c) = [ ci+j−2 ]ni,j=1 .

(7.4)

we have Diﬀerentiating the equality (7.3) we obtain Ln (c, 2) = [ (i + j − 2)ci+j−3 ]ni,j=1 and in general, for m = 1, . . . , 2n − 1,

n i+j−2 i+j−1−m c . Ln (c, m) = m−1 i,j=1 It is obvious that the rank of Ln (c, m) is equal to m. The matrices Ln (c, m) are called elementary Hankel matrices. Example 7.2. For our second example we assume that F is algebraically closed. Let u(t) be a polynomial of degree n and let t1 , . . . , tn be the zeros of u(t). The Newton sums si (i = 1, 2, . . . ) are given by si =

n

ti−1 . k

k=1

We form the Hankel matrix Hn = [ si+j−1 ]ni,j=1 . Then we have u (t) Hn = H n . u(t) This follows from the obvious relation u (t) 1 . = u(t) t − tk n

k=1

The transformation H : f (t) −→ Hn (f ) is clearly a linear operator from the vector space of all proper rational functions to the space of n × n Hankel matrices. The kernel of this operator consists of all proper rational function f (t) for which f (t) − f (∞) = O(t−2n ). We show that this transformation is onto. That means any k × k Hankel matrix can be regarded as generated by a proper rational function.

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Proposition 7.3. Let u(t) be a ﬁxed monic polynomial of degree 2n − 1. Then any n × n Hankel matrix Hn can be represented uniquely in the form p Hn = H n (7.5) u for some p ∈ F2n−1 . Proof. Clearly, a matrix of the form (7.5) with p ∈ F2n−1 does not belong to the kernel of the transformation H. That means that the mapping of the vector p ∈ F2n−1 to the Hankel matrix Hn p u is one-to-one. Hence the dimension of its range equals 2n − 1. This is just the dimension of the space of n × n Hankel matrices. Thus the mapping is onto. Since in an algebraically closed ﬁeld F any proper rational function has a partial fraction decomposition we conclude that in this case any Hankel matrix can be represented as a linear combination of elementary Hankel matrices. The reader may observe that the problem to ﬁnd the generating function of a Hankel matrix is closely related to the Pad´e approximation problem at inﬁnity and the partial realization problem. In connection with these and other problems the question about a generating function of minimal degree arises. We will see later that the degree of the generating function is at least equal to the rank of Hn . But it can be bigger. For example, the rank-one Hankel matrix Hn = en eTn has no generating function of degree less than 2n − 1. Here we restrict ourselves to the nonsingular case. For a nonsingular n × n Hankel matrix, a generating function of degree n always exists, as the next proposition shows. Let us use the notation of Section 4.1. Theorem 7.4. Let Hn = [ si+j−1 ]ni,j=1 be a nonsingular Hankel matrix, {u(t), v(t)} be a fundamental system of Hn , where u(t) is monic and deg v(t) < n. Then, for any α ∈ F , there is a vector p ∈ Fn such that

p(t) . Hn = Hn u(t) − αv(t) Proof. We consider the (n − 1) × (n + 1) matrix ∂Hn , which was introduced in (4.1). The vector w = u − αv is a monic vector belonging to the nullspace of ∂Hn . We set ⎡ ⎤ 0 s1 . . . sn ⎢ .. ⎥ w . .. .. p=⎣ . . . ⎦ 0

s1 From this deﬁnition and from w ∈ ker ∂Hn we can see that in the expansion p(t) = h1 t−1 + h2 t−2 + · · · w(t) p(t) we have hi = si , i = 1, . . . , 2n − 1 . Hence, Hn = Hn w(t) .

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Example 7.5. Let us ﬁnd generating functions of degree n of the matrix Hn = Jn . For this matrix {en+1 , e1 } is a fundamental system. Furthermore, p = e1 . Thus, for any α ∈ F,

1 Jn = Hn n . t −α Let us present a special case of Theorem 7.4 involving the solutions of the equations Hk uk = ρk ek , eTk uk = 1 (7.6) for k = n, n + 1 . Here Hn+1 is a nonsingular extension of Hn . As we already know from Section 4.2 these monic solutions form a fundamental system for Hn . Thus, the following is immediately clear. Corollary 7.6. Let Hn and Hn+1 be as above and un , un+1 be the solutions of (7.6) for k = n, n + 1 . Then, for an α ∈ F , there are vectors pn+1 ∈ Fn and pn ∈ Fn−1 such that

pn+1 (t) − αpn (t) . Hn = Hn un+1 (t) − αun (t) 2. Vandermonde factorization of Hankel matrices. In this subsection we assume that F = C. Let Hn be an n × n nonsingular Hankel matrix. Then, by Theorem 7.4, it has a generating function of degree n. We can assume that the denominator polynomial of this function has only simple roots, which follows from the following lemma. Recall from Corollary 4.3 that the polynomials forming a fundamental system of a nonsingular Hankel matrix are coprime. Lemma 7.7. Let u(t) and v(t) be coprime. Then for all α ∈ C, with the exception of a ﬁnite number of points, the polynomial w(t) = u(t) − αv(t) has only simple roots. Proof. Suppose that τ0 is a multiple root of w(t). Then u(τ0 ) = αv(τ0 ) and u (τ0 ) = αv (τ0 ). Since u(t) and v(t) are coprime, v(τ0 ) = 0. Hence τ0 is a root of the nonzero polynomial z(t) = u(t)v (t) − v(t)u (t). Now we choose α such that u(τ ) = αv(τ ) for all roots τ of z(t). Then none of the τ is a root of w(t), so w(t) has no multiple roots. p(t) Let f (t) = w(t) with (∞) = 0 be a proper rational function in reduced form such that w(t) has simple roots t1 , . . . , tn . Then f (t) has a partial fraction decomposition n δi f (t) = , t − ti i=1

where p(ti ) δi = = w (ti )

−1 1 (ti ) . f

(7.7)

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Hence Hk (f ) can be represented as a linear combination of elementary Hankel matrices Lk (ti ) deﬁned by (7.4) Hk (f ) =

n

δi Lk (ti ) .

(7.8)

i=1

This relation can be stated as a matrix factorization. In fact, observe that Lk (ti ) equals k (ti )k (ti )T , where k (ti ) = ( tj−1 )kj=1 . We form the n × k Vandermonde i n matrix Vk (t), t = (ti )i=1 , with the rows k (ti )T (i = 1, . . . , n, ) k ]ni=1,j=1 . Vk (t) = [ tj−1 i

Now (7.8) is equivalent to the following. Proposition 7.8. Let f (t) and t be as above. Then for k ≥ n, the Hankel matrix Hk = Hk (f ) admits a representation Hk = Vk (t)T DVk (t) , where D =

diag (δi )ni=1

(7.9)

and the δi are given by (7.7).

Example 7.9. For the Hankel matrix of Example 7.2 we obtain the following factorization, u = Vn ( t)T diag (νi )ri=1 Vn ( t) , (7.10) Hn u where t is the vector of diﬀerent zeros ti (i = 1, . . . , r) of u(t), and νi are their multiplicities. A consequence of this Proposition 7.8 is that rank Hk (f ) = n for all k ≥ n. In Section 7.5 we will show that this is true for a general ﬁeld F. Combining Proposition 7.8 with Proposition 7.4 we obtain the following. Corollary 7.10. Let Hn be a nonsingular n×n Hankel matrix, {u, v} a fundamental system of Hn , and α ∈ C such that w(t) = u(t)−αv(t) has simple roots t1 , . . . , tn . Then Hn admits a representation Hn = Vn (t)T DVn (t) with a diagonal matrix D. Let us ﬁnd the Vandermonde factorization for Example 7.5, i.e., for Hn = Jn . The polynomial tn − α has simple roots for all α = 0, and these roots ti are the nth complex roots of α. The diagonal matrix is given by 1 D = diag (t1−n , . . . , t1−n ). n 1 n 3. Real Hankel matrices. We consider the special case of a real, nonsingular Hankel matrix Hn . In this case the fundamental system of Hn is also real. We choose α ∈ R, since then the non-real roots of w(t) = u(t) − αv(t) appear in conjugate complex

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pairs. Let t1 , . . . , tr be the real roots and tr+1 , tr+2 = tr+1 , . . . , tn−1 , tn = tn−1 be the non-real roots of w(t). Then Vn (t)T = Vn (t)∗ diag(Ir , J2 , . . . , J2 ) , l

where r + 2l = n. Thus, we obtain from Proposition 7.8 the following. Corollary 7.11. If the Hankel matrix Hk (k ≥ n) in Proposition 7.8 is real, then it admits a representation Hk = Vk (t)∗ D1 Vk (t) , where

D1 = diag δ1 , . . . , δr ,

0 δr+1

δ r+1 0

,...,

0 δn−1

δ n−1 0

.

In particular, the matrices Hn and D1 are congruent. Combining this with Sylvester’s inertia law, with (7.9), and with the fact that 0 δ equals 0, we conclude the following. the signature of a matrix δ 0 Corollary 7.12. Let f (t) be a real rational function of degree n with simple poles t1 , . . . , tn , where t1 , . . . , tr are real and the other poles non-real. Then for k ≥ n the signature of the Hankel matrix Hk = Hk (f ) is given by r sgn δi , sgn Hk = i=1

where δi is deﬁned by (7.7). In particular, Hn is positive deﬁnite if and only if all roots of w(t) are real and δi > 0 for all i. Let us specify the criterion of positive deﬁniteness further. 4. The Cauchy index. Let C be an oriented closed curve in the extended complex plane C∪{ ∞ } and f (t) a rational function with real values on C with the exception of poles. A pole c of f (t) on C is said to be of positive type if lim f (t) = −∞ and

t→c− t∈C

lim f (t) = ∞ .

t→c+ t∈C

It is said to be of negative type if c is of positive type for −f (t). If a pole is not of positive or negative type, then it is called neutral. The Cauchy index of f (t) along C is, by deﬁnition, the integer indC f (t) = p+ − p− where p+ is the number of poles of positive and p− the number of poles of negative 1 type. The pole c is of positive (negative) type if and only if the function f (t) is increasing (decreasing) in a neighborhood of c . It is clear that if c is a pole of positive or negative type on C, then a small perturbation of the coeﬃcients of f (t) leads only to a small change of the pole on

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C by preserving its type (which is not true for neutral poles). Now we are in the position to relate the signature of Hankel matrices generated by a rational function with the Cauchy index of this function along R. Proposition 7.13. Let f (t) be a real proper rational function of degree n . Then sgn Hn (f ) = indR f (t) . is the reduced representation as quotient of polyProof. Suppose that f (t) = p(t) u(t) nomials. Let us ﬁrst assume that u(t) has simple roots t1 , . . . , tn , i.e., f (t) has simple Simple poles cannot be neutral. If ti is a simple pole of positive type, poles. then 1f (ti ) > 0. Comparing this with (7.7) we conclude that δi > 0. Analogously, we have δi < 0 for a pole ti of negative type. Now it remains to apply Corollary 7.12. Now let u(t) have multiple roots. Neutral poles of f (t) correspond to roots of u(t) of even order and do not contribute to the Cauchy index of f (t) . It is easy to see that we can disturb u(t) additively with an α ∈ R as small as we want such that the disturbed roots of even order disappear or become simple roots of u(t) + α , so that the respective pairs of poles do not contribute to the Cauchy p(t) index of fα (t) = u(t)+α . The other poles remain simple and of the same type. So the ﬁrst part of the proof applies to fα (t) . Due to Proposition 7.17 below Hn (f ) is nonsingular. Taking into account that the signature of a nonsingular Hankel matrix is invariant with respect to small perturbations the assertion follows. Some readers might be unsatisﬁed with the analytic argument in the proof of the algebraic Proposition 7.13. For those readers we note that a purely algebraic proof of this proposition is possible if more general Vandermonde representations of Hankel matrices are considered. Let us discuss the question how positive deﬁniteness of Hn (f ) can be characterized in terms of f (t). According to Proposition 7.13, Hn (f ) is positive deﬁnite if and only if the Cauchy index of f (t) along R is equal to n. That means that f (t) must have n poles of positive type. Between two poles of positive type there must be a zero of f (t), i.e., a root of the numerator polynomial. In other words, the poles and zeros of f (t) must interlace. We say that the real roots of two polynomials u(t) and p(t) interlace if between two roots of u(t) there is exactly one root of p(t). Polynomials with roots that interlace are coprime. Let us summarize our discussion. Corollary 7.14. Let f (t) be a real rational function of degree n, and let f (t) = p(t) u(t) be its reduced representation. Then Hn (f ) is positive deﬁnite if and only if the polynomials u(t) and p(t) have only real simple roots that interlace. 5. Congruence to H-Bezoutians. In Section 4 we showed that inverses of Hankel matrices are H-Bezoutians. Now we are going to explain another relation between Hankel matrices and H-Bezoutians. In the real case this relation just means that Hankel matrices and H-Bezoutians are congruent.

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Throughout this subsection, let u(t) be a polynomial of degree n, v(t) ∈ v(t) . Then B = BezH (u, v) is an n × n matrix. For k > n, Fn+1 (t), and f (t) = u(t) we identify this matrix with the k × k matrix obtained from B by adding k − n zero rows and zero columns at the bottom and on the right. The same we do for B(u) = BezH (u, e1 ) introduced in (2.2). Proposition 7.15. For k ≥ n, the k × k Hankel matrix generated by f (t) = related to the H-Bezoutian of u(t) and v(t) via BezH (u, v) = B(u) Hk (f ) B(u) .

v(t) u(t)

is

(7.11)

Proof. For B = BezH (u, v) and u = (ui )n+1 i=1 we have in view of (7.2) B(t, s) = =

=

f (t) − f (s) u(s) t−s n+1 ∞ ui hp+q−1 uj ti−p−1 sj−q−1

−u(t)

i,j=1 p,q=1 ∞ n

um+p hp+q−1 uq+l tm−1 sl−1 ,

m,l=1 p,q=1

where we set uj = 0 for j > n + 1. The coeﬃcient matrix of this polynomial can be written as a product of three matrices, as it is claimed. Recall from Corollary 3.4 that the nullity of BezH (u, v) is equal to the degree of the greatest common divisor of u(t) and v(t). This implies that the rank of BezH (u, v) is equal to the degree of the rational function f (t). Corollary 7.16. In the real case the matrices BezH (u, v) and Hn uv are congruent. Proposition 7.17. Let f (t) be a rational function of degree n. Then, for k ≥ n, the rank of Hk (f ) is equal to n. In particular, Hn (f ) is nonsingular. Proof. Let f (t) = v(t) u(t) be in reduced form and deg u(t) = n . Since the matrix B(u) in (7.11) as an k × k matrix is singular for k > n , the assertion cannot be concluded directly from Proposition 7.15. Thus, for k > n , deﬁne uk and vk by uk (t) = tk u(t) and vk (t) = tk v(t) , respectively. The f (t) = uvkk (t) (t) and, due to Proposition 7.15, (7.12) BezH (uk , vk ) = B(uk ) Hk (f ) B(uk ) , where B(uk ) is nonsingular. Since the nullity of BezH (uk , vk ) equals k , we have that rank BezH (uk , vk ) = n = rank Hk (f ) . Note that under the assumptions of Proposition 7.8 the assertion of Proposition 7.17 follows already from the Vandermonde factorization of Hk (f ) given in (7.9). In the real case, (7.12) is a congruence relation. Applying Sylvester’s inertia law we can conclude the following for the case F = R.

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n+1 (t), then Proposition 7.18. If u(t) ∈ Rn+1 (t) with deg u(t) = n and v(t) v∈ R the matrices BezH (u, v) (considered as k × k matrix) and Hk u have the same inertia. v(t) Proof. Due to formula (7.12), which is also true if u(t) is not in reduced form, and since BezH (u, v) and BezH (uk , vk ) have the same rank it remains to show that these two Bezoutians have the same signature. But, this follows from

BezH (uk , vk ) = Mn (ek+1 )BezH (u, v)Mn (ek+1 )T

and from Corollary 1.2.

6. Inverses of H-Bezoutians. Comparing (7.11) with Barnett’s formula (6.3) we obtain that the Hankel matrix Hn p admits a representation u p (7.13) = p(C(u))B(u)−1 . Hn u Together with (6.6) this immediately leads to the following. Theorem 7.19. Let u(t) and v(t) be coprime and (q(t), p(t)), be the (unique) solution of the Bezout equation u(t)q(t) + v(t)p(t) = 1 with q(t), p(t) ∈ F (t). Then

(7.14)

n

BezH (u, v)−1 = Hn

p

. u An immediate consequence of this theorem is the converse of Theorem 4.2. Corollary 7.20. The inverse of a nonsingular H-Bezoutian is a Hankel matrix. Furthermore, Theorem 7.19 tells us that the inverse of a Bezoutian or the inverse of a Hankel matrix generated by a rational function can be computed with an algorithm which solves the Bezout equation (7.14). We show next that this equation can be solved with the help of the Euclidian algorithm. 7. Solving the Bezout equation. Let ui (t) be the polynomials computed by the Euclidian algorithm, as described in Section 5.3, then with the help of the data of the Euclidian algorithm we can recursively solve the Bezout equations u(t)xi (t) + v(t)yi (t) = ui (t)

(i = 0, 1, . . . ),

(7.15)

where, for initialization, we have x0 (t) = y0 (t) =

1 , 0

x1 (t) y1 (t)

= 0 . = 1

The recursion is given by xi+1 (t) =

xi−1 (t) − qi (t)xi (t)

yi+1 (t) =

yi−1 (t) − qi (t)yi (t) .

(7.16)

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77

In fact, u(t)xi+1 (t) + v(t)yi+1 (t) = u(t)(xi−1 − qi (t)xi (t)) + v(t)(yi−1 (t) − qi (t)yi (t)) = ui−1 (t) − qi (t)ui (t) = ui+1 (t) . Introducing the 2 × 2 matrix polynomials

xi−1 (t) xi (t) 0 1 and Φi (t) = Xi (t) = yi−1 (t) yi (t) 1 −qi (t) we can write the recursion (7.16) as Xi+1 (t) = Xi (t)Φi (t) ,

X1 (t) = I2 .

If u(t) and v(t) are coprime, then we have for some i = l that ul (t) = c = const. Thus, the solution of (7.14) is obtained from (xl (t), yl (t)) by dividing by c. As mentioned in Section 5.3 the Euclidian algorithm for ﬁnding the greatest common divisor of two polynomials is closely related to a Schur-type algorithm for Hankel matrices. The Euclidian algorithm for solving the Bezout equation is related to a mixed Levinson-Schur-type algorithm for Hankel matrices. It is also possible to design a pure Levinson-type algorithm for the solution of the Bezout equation.

8. Toeplitz matrices generated by rational functions This section is the Toeplitz counterpart of the previous one. We deﬁne and study Toeplitz matrices generated by rational functions and show that they are closely related to T-Bezoutians. Some features are completely analogous to the Hankel case, but there are also some signiﬁcant diﬀerences. 1. Generating functions of Toeplitz matrices. Let f (t) = p(t) u(t) be a proper rational function with u(0) = 0. Then f (t) admits series expansions in powers of t and as well as in powers of t−1 , 2 f (t) = a+ 0 + a1 t + a2 t + · · · ,

−1 f (t) = −a− − a−2 t−2 − · · · . 0 − a−1 t

(8.1)

If F = C, then (8.1) can be understood as the Laurent series expansion at t = 0 and at t = ∞, respectively. For a general F, (8.1) makes sense as the quotient of two formal Laurent series. The coeﬃcients can be obtained recursively by obvious relations. Note that, in a formal sense, ∞ f (t) − f (s−1 ) = ai−j ti sj , (8.2) 1 − ts i,j=0 − where a0 = a+ 0 + a0 . The latter follows from the obvious relation

(1 − ts)T (t, s) = a0 +

∞ i=1

where T =

[ ai−j ]∞ i,j=1 .

i

ai t +

∞

a−j sj ,

j=1

This relation makes the following deﬁnition natural.

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For n = 1, 2, . . . the n × n Toeplitz matrix generated by f (t) with the expan− sions (8.1) is, by deﬁnition, the matrix Tn(f ) = [ ai−j ]ni,j=1 , where a0 = a+ 0 + a0 . If Tn = Tn (f ), then the function f (t) is called generating function of Tn . Obviously, Tn (f ) is the zero matrix if f is a constant function. Note that ﬁnding the generating function of a given Toeplitz matrix is a two-point Pad´e approximation problem for the points 0 and ∞. Example 8.1. Since for c = 0 ∞

∞

1 = ck tk 1 − ct

1 =− c−k t−k , 1 − ct

and

k=0

we have

Tn

1 1 − ct

k=1

T = [ ci−j ]ni,j=1 = n (c)n c−1 ,

(8.3)

where n (c) is deﬁned in (1.1). Example 8.2. Our second example is the Toeplitz analogue of Example 7.2. Let F be algebraically closed and u(t) a polynomial of degree n with the roots t1 , . . . , tn and u(0) = 0. We deﬁne ci =

n

tik

(i = 0, ±1, ±2, . . . )

k=1

and form the Toeplitz matrix Tn = [ ci−j ]ni,j=1 . Then Tn = Tn (f ) for f (t) =

n k=1

J

Taking u (t) =

n k=1 (1

1 . 1 − tk t

− tk t) into account we ﬁnd that f (t) =

(u )J (t) . uJ (t)

Here u has to be considered as a vector in Fn , that means (u )J (t) = tn−1 u (t−1 ) . Like for Hankel matrices, it can be shown that, for any given polynomial u(t) of degree 2n−2 with u(0) = 0 , any n×n Toeplitz matrix has a generating function with denominator polynomial u(t). More important is the following Toeplitz analogue of Theorem 7.4 about generating functions of nonsingular Hankel matrices. Note that its proof is somehow diﬀerent to the Hankel case. Theorem 8.3. Let Tn = [ ai−j ]ni,j=1 be a nonsingular Toeplitz matrix, {u(t), v(t)} be a fundamental system of Tn . Furthermore, let α, β ∈ F be such that w(t) = αu(t) + βv(t) is of degree n and w(0) = 0. Then there is a p ∈ Fn+1 such that p Tn = Tn . w

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79

Proof. Let ∂Tn be the matrix deﬁned in (4.7). Then w belongs to the nullspace of ∂Tn . We ﬁnd an and a−n via the equations an an−1 . . . a0 w = a0 . . . a1−n a−n w = 0 and form the (n + 1) × (n + 1) Toeplitz matrix Tn+1 = [ ai−j ]n+1 i,j=1 . Then we have + − + Tn+1 w = 0. Now we represent Tn+1 as Tn+1 = Tn+1 + Tn+1 , where Tn+1 is a lower − triangular and Tn+1 is an upper triangular Toeplitz matrix and deﬁne + p = Tn+1 w.

(8.4)

− w. A comparison of coeﬃcients reveals that We have also p = −Tn+1

p(t) n = a+ 0 +a1 t+· · ·+an t +· · · w(t)

and

p(t) −1 −· · ·−a−n t−n −· · · , = −a− 0 −a−1 t w(t)

± where a± 0 are the diagonal entries of Tn+1 , respectively. Thus Tn = Tn (f ) for p(t) f (t) = w(t) .

Example 8.4. Let us compute generating functions for the identity matrix In . We observe ﬁrst that {e1 , en+1 } is a fundamental system. To meet the conditions of the theorem we choose α = 0, β = 0. Then we ﬁnd that an = − αβ and a−n = − α β. We can choose now β + Tn+1 = γIn+1 − en+1 eT1 , α where γ is arbitrary. For γ = 0 this leads to p = −βen+1 . Thus generating functions for In are given by −βtn . f (t) = α + βtn If we choose a diﬀerent γ, then the resulting function diﬀers from that function only by a constant. 2. Matrices with symmetry properties. It is a little bit surprising that if f (t) is symmetric in the sense that f (t−1 ) = f (t) , (8.5) then the matrix Tn (f ) becomes skewsymmetric. Symmetric matrices Tn (f ) are obtained if f (t) satisﬁes f (t−1 ) = −f (t) . (8.6) In the case F = C the matrices Tn (f ) are Hermitian if f( t

−1

) = −f (t) .

(8.7)

This is equivalent to saying that f (t) takes purely imaginary values on the unit circle. We show that the converse is, in a sense, also true. Proposition 8.5. If Tn is a nonsingular symmetric, skewsymmetric or Hermitian Toeplitz matrix, then there exists a generating function f (t) for Tn of degree n that satisﬁes the conditions (8.6), (8.5), (8.7), respectively.

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Proof. If Tn is symmetric, then the fundamental system consists of a symmetric vector u and a skewsymmetric vector v. If the last component of u does not vanish, then we can choose w = u to satisfy the conditions of Theorem 8.3. Further, we − + obtain a−n = an , thus Tn+1 is symmetric, and the choice Tn+1 = (Tn+1 )T is possible. Hence we have + + + p = Tn+1 u = −(Tn+1 )T u = −Tn+1 uJ = −pJ .

That means that p is skewsymmetric, which implies that f (t) = p(t) u(t) satisﬁes (8.6). If the last component of u vanishes, then the last component of v must be nonzero and we can choose w = v. Again we obtain a−n = an , so that Tn+1 is − + symmetric. With the choice Tn+1 = (Tn+1 )T we have + + + p = Tn+1 v = −(Tn+1 )T v = −Tn+1 vJ = pJ .

satisﬁes (8.6). The proof of the other cases is Thus, p is symmetric and f (t) = p(t) v(t) analogous. We have to take into account that a fundamental system of a nonsingular skewsymmetric Toeplitz matrix consists of two symmetric vectors and the fundamental system of a nonsingular Hermitian Toeplitz matrix of two conjugatesymmetric vectors. To discuss this proposition we consider the Examples 8.1, 8.4. The Toeplitz matrix Tn = [ ci−j ]ni,j=1 in Example 8.1 is Hermitian if c is on the unit circle, but its 1 1 1+ct does not satisfy (8.7). However, since 1−ct = 12 1−ct +1 generating function 1−ct we have also

1 1 + ct , T n = Tn 2 1 − ct and this generating function satisﬁes (8.7). Generating functions for the identity matrix In satisfying (8.6) and so reﬂecting its symmetry are f (t) =

1 1 − tn 2 1 + tn

and f (t) =

1 1 + tn . 2 1 − tn

3. Vandermonde factorization of nonsingular Toeplitz matrices. Let Tn be a nonp(t) be a generating singular n×n Toeplitz matrix with complex entries and f (t) = w(t) function of degree n with f (∞) = 0 . According to Theorem 8.3 and Lemma 7.7 such a function exists and, due to the freedom in the choice of w(t) we can assume that w(t) has only simple roots t1 , . . . , tn . Using the partial fraction decomposition of f (t) in the form n 1 γi f (t) = − ti 1 − t−1 i t i=1 as well as (8.3) we can conclude, in analogy to the Hankel case, the following. Proposition 8.6. Let Tn be a nonsingular n × n Toeplitz matrix, {u, v} a fundamental system of Tn and α, β ∈ C such that w(t) = αu(t) + βv(t) is of degree n,

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satisﬁes w(0) = 0, and has simple roots t1 , . . . , tn . Then Tn admits a representation Tn = Vn (t−1 )T DVn (t) (8.8) −1 n −1 n n −1 where t = (ti )i=1 and t = (ti )i=1 , and D is diagonal, D = diag −tk γk k=1 . The diagonal matrix can be expressed in terms of the generating function. If D = diag (δi )ni=1 , then −1 1 1 1 p(ti ) =− δi = − (ti ) . (8.9) ti w (ti ) ti f Note that, like for Hankel matrices, the Vandermonde factorization for a nonsingular Toeplitz matrix Tn (f ) extends to all Toeplitz matrices Tk (f ) with k ≥ n as (8.10) Tk (f ) = Vk (t−1 )T DVk (t) . 4. Hermitian Toeplitz matrices. Let the assumptions of Proposition 8.6 be satisﬁed. We consider the special case of an Hermitian Toeplitz matrix Tn . In this case Tn has a fundamental system consisting of two conjugate-symmetric vectors. If we choose α and β as reals, then the vector w in Proposition 8.6 is also conjugatesymmetric. For a polynomial with a conjugate-symmetric coeﬃcient vector the roots are symmetric with respect to the unit circle T. That means if t0 is a root, −1 then t0 is also a root. In particular, the roots not on T appear in pairs that are symmetric with respect to T. −1 Let t1 , . . . , tr be the roots of w(t) on T and tr+1 , tr+2 = tr+1 , . . . , tn−1 , tn = −1 tn−1 be the roots of w(t) which are not on T. Note that for the coeﬃcients γi in the partial fraction decomposition of f (t), which are the residuals at ti , we have δr+2 = δ r+1 , . . . , δn = δ n−1 . Furthermore, Vn (t−1 )T = Vn (t)∗ diag(Ir , J2 , . . . , J2 ) , l

where r + 2l = n. Now Proposition 8.6 leads to the following. Corollary 8.7. If the Toeplitz matrix in Proposition 8.6 is Hermitian, then it admits a representation Tn = Vn (t)∗ D1 Vn (t) , where

δ r+1 δ n−1 0 0 D1 = diag δ1 , . . . , δr , ,..., δr+1 0 δn−1 0 In particular, the matrices Tn and D1 are congruent. 5. Signature and Cauchy index. Corollary 8.7 allows us to express the signature of Tn in terms of the signs of the diagonal elements δi (i = 1, . . . , r) of D1 (cf. Corollary 7.12). Our aim is now to express it in terms of the Cauchy index.

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Let f (t) be a rational function of degree n satisfying (8.7). Then the function f (t) takes real values on the unit circle, thus the Cauchy index (see Section 7.4) indT 1i f (t) is well deﬁned. It is the diﬀerence of the number of poles on T of positive type and the number of poles of negative type. Let tj = eiθj be a pole on T. Then tj is of positive (negative) type if and only if the (real-valued) function 1 i

ϕ(θ) =

i f (eiθ )

is increasing (decreasing) in a neighborhood of θj . For a simple pole this is equivalent to ϕ (θj ) > 0 (ϕ (θj ) < 0). We have 1 ϕ (θj ) = −tj (tj ) . f Comparing this with (8.9) we conclude that ϕ(θj ) = δj−1 . We arrive at the following statement for the case of simple poles. It can be generalized to multiple poles by using the continuity arguments from the proof of Proposition 7.13. Proposition 8.8. Let f (t) be a proper rational function with degree n that takes imaginary values on the unit circle. Then 1 sgn Tn (f ) = indT f (t) . i We also obtain a criterion of positive deﬁniteness. Corollary 8.9. Let f (t) be a proper rational function of degree n, and let f (t) =

ip(t) u(t)

be its reduced representation in which u and p are conjugate-symmetric. Then Tn (f ) is positive deﬁnite if and only if the polynomials u(t) and p(t) have only roots on the unit circle, these roots are simple and interlaced. 6. Congruence to T-Bezoutians. The Toeplitz analogue of Proposition 7.15 is as follows. (Concerning the order of BezT (u, v) and of B± (u) compare the remarks before Proposition 7.15.) Proposition 8.10. Let u, v ∈ Fn+1 , where u has nonvanishing ﬁrst and last components. Then, for k ≥ n, the T-Bezoutian of u(t) and v(t) is related to the k × k Toeplitz matrix Tk (f ) generated by f (t) = v(t) u(t) via BezT (u, v) = B− (u) Tk (f ) B+ (u) , where B± (u) are introduced in (2.14). Proof. For B = BezT (u, v) we have B(t, s) = u(t)

f (t) − f (s−1 ) (−uJ (s)) . 1 − ts

Introduction to Bezoutians Applying (8.2) we obtain ⎤

⎡

u1 ⎢ .. B=⎣ . un

..

. ...

u1

⎡

⎢ ⎥ ⎢ T ⎢ ⎦ O ⎣

−un+1

... .. .

83

−u2 .. . −un+1

⎤ ⎥ ⎥ ⎥ , ⎦

O

∞ where u = (ui )n+1 i=1 and T = [ ai−j ]i,j=1 . Hence B = B− (u)Tk (f )B+ (u) .

Corollary 8.11. If Tn vu is Hermitian and if u is a conjugate-symmetric (or conjugate-skewsymmetric) polynomial of degree n + 1 then BezT (v, u) (or BezT (u, v)) and Tn (f ) are congruent. Corollary 8.12. For k ≥ n the rank of Tk (f ) is equal to the degree of f (t). In particular, if f (t) has degree n, then Tn (f ) is nonsingular. If we combine Proposition 8.10 with Barnett’s formula in Theorem 6.3, then we obtain the representation v Tn (8.11) = vJ (C(uJ ))B+ (u)−1 , u where we assume that u(t) is comonic. 7. Inverses of T-Bezoutians. Now we show that relation (8.11) leads to an inversion formula for T-Bezoutians. Theorem 8.13. Let u, v ∈ Fn+1 be such that u(t) and v(t) are coprime and the ﬁrst and last components of u do not vanish. If (q(t), p(t)), q, p ∈ Fn+1 , is the solution of the Diophantine equation u(t)q(t) + v(t)p(t) = tn ,

(8.12)

then BezT (u, v) is invertible and the inverse is given by p . BezT (u, v)−1 = Tn u Proof. First we note that (8.12) is equivalent to uJ (t)qJ (t) + vJ (t)pJ (t) = tn . Thus, vJ (t)pJ (t) ≡ tn modulo uJ (t). From Theorem 6.3, the representation (8.11), and the Cayley-Hamilton theorem, now we obtain p = B− (u)vJ (C(uJ ))pJ (C(uJ ))B+ (u)−1 BezT (u, v)Tn u = B− (u)C(uJ )n B+ (u)−1 . Taking (6.7) into account this leads to p BezT (u, v)Tn = B+ (u)B+ (u)−1 = In , u and the theorem is proved.

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8. Relations between Toeplitz and Hankel matrices. We know that if Tk is a k × k Toeplitz matrix, then Jk Tk is Hankel, and vice versa. We show how the generating functions are related. Proposition 8.14. Let u, q ∈ Fn+1 , where u has nonvanishing ﬁrst and last comq(t) ponents, and let u(t) be a generating function of Tk . For k ≥ n , let p ∈ Fn be such n that −p(t) ∈ F (t) is the remainder polynomial of tk qJ (t) divided by uJ (t) . Then q p = Hk . Jk T k u u Proof. According to the deﬁnition of p we have tk qJ (t) = −pJ (t) + r(t)uJ (t) for some r(t) ∈ Fk+1 (t). This is equivalent to q(t) = −tk p(t) + tk r(t−1 )u(t) , and we obtain p(t) q(t) = −tk + tk r(t−1 ). u(t) u(t) Thus k tk r(t−1 ) = a+ 0 + a 1 t + · · · + ak t . On the other hand, q(t) p(t) = r(t−1 ) − t−k . u(t) u(t) Consequently, p(t) = ak + ak−1 t−1 + · · · + a0 t−k a−1 t−k−1 + · · · , u(t) − where a0 = a+ 0 + a0 . This means that ⎡ ak−1 p ⎢ ak−2 ⎢ Hk = Hk =⎢ . u ⎣ ..

Thus Hk = Jk Tk

q u

a0

ak−2 ..

. ...

.

... . ..

a0 .. .

⎤ ⎥ ⎥ ⎥ . ⎦

a1−k

9. Vandermonde reduction of Bezoutians In this section the underlying ﬁeld is the ﬁeld of complex numbers, F = C. Some of the results can be extended to general algebraically closed ﬁelds. In Sections 7 and 8 we showed that nonsingular Hankel and Toeplitz matrices can be represented as a product of the transpose of a Vandermonde matrix VnT , a diagonal matrix, and the Vandermonde matrix Vn , and we called this Vandermonde factorization. Since inverses of Hankel and Toeplitz matrices are H- and TBezoutians, respectively, this is equivalent to the fact that nonsingular Bezoutians can be reduced to diagonal form by multiplying them by Vn from the left and by

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85

VnT from the right. We call this kind of factorization Vandermonde reduction of Bezoutians. In this section we give a direct derivation of Vandermonde reduction of Bezoutians and generalize it to general, not necessarily nonsingular Bezoutians. Hereafter, the notations conﬂuent and non-conﬂuent are used in connection with conﬂuent and non-conﬂuent Vandermonde matrices, respectively. 1. Non-conﬂuent Hankel case. To begin with, let us recall that, for t = (ti )ni=1 , the (non-conﬂuent) Vandermonde matrix Vm (t) is deﬁned by ⎤ ⎡ 1 t1 . . . tm−1 1 ⎢ 1 t2 . . . tm−1 ⎥ 2 ⎥ ⎢ Vm (t) = ⎢ . . .. ⎥ . . . ⎣ . . . ⎦ 1 tn

...

tm−1 n

If m = n and the ti are distinct, then Vn (t) is nonsingular. Obviously, for x ∈ Cm , Vm (t)x = (x(ti ))ni=1 . Furthermore, for an n×n matrix B and s = (sj )nj=1 , Vn (t)BVn (s)T = [ B(ti , sj ) ]ni,j=1 .

(9.1)

We specify this for t = s with pairwise distinct components and an H-Bezoutian B = BezH (u, v), where u(t), v(t) ∈ Cn+1 (t) and u(t) has degree n . In this case the oﬀ diagonal entries cij of C = Vn (t)BVn (t)T are given by cij =

u(ti )v(tj ) − v(ti )u(tj ) . ti − t j

(9.2)

Our aim is to ﬁnd t such that C is diagonal. One possibility is to choose the zeros of u(t). This is possible if u(t) has only simple zeros. A more general case is presented next. Proposition 9.1. Let α ∈ C be such that w(t) = u(t) − αv(t) has simple roots t1 , . . . , tn , and let t = (ti )ni=1 . Then Vn (t)BezH (u, v)Vn (t)T = diag(γi )ni=1 ,

(9.3)

γi = u (ti )v(ti ) − u(ti )v (ti ) .

(9.4)

where Proof. According to Lemma 2.2 we have BezH (u, v) = BezH (w, v). From (9.2) we see that the oﬀ diagonal elements of the matrix Vn (t)BezH (u, v)Vn (t)T vanish. )−v(t)u(ti ) The expression (9.4) follows from (9.2), γi = lim u(t)v(tit−t . i t→ti

Remark 9.2. If u(t) and v(t) are coprime, which is the same as saying that BezH (u, v) is nonsingular, then according to Lemma 7.7 for almost all values of α the polynomial w(t) has simple roots. Remarkably the special case v(t) = 1 of (9.3) leads to a conclusion concerning the inverse of a Vandermonde matrix.

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Corollary 9.3. Let t = (ti )ni=1 have pairwise distinct components, and let u(t) be deﬁned by u(t) = ni=1 (t − ti ). Then the inverse of Vn (t) is given by

n 1 −1 T Vn (t) = B(u)Vn (t) diag , (9.5) u (ti ) i=1 where B(u) is the upper triangular Hankel matrix introduced in (2.2). Now we consider the H-Bezoutian of real polynomials u(t) and v(t), which is an Hermitian matrix. Similarly to Corollary 7.11, Proposition 9.1 leads to a matrix congruence. Corollary 9.4. Let in Proposition 9.1 the polynomials u(t) and v(t) be real and α ∈ R. Furthermore, let t1 , . . . , tr be the (simple) real and (tr+1 , tr+1 ), . . . , (tn−1 , tn−1 ) the (simple) non-real roots of w(t). Then

γ r+1 γ n−1 0 0 ∗ ,..., . Vn (t)BezH (u, v)Vn (t) = diag γ1 , . . . , γr , γr+1 0 γn−1 0 In particular, sgn BezH (u, v) =

r

sgn γi .

i=1

Note that it follows from the Hermitian symmetry of the matrix that the numbers γ1 , . . . , γr are real. 2. Non-conﬂuent Toeplitz case. Let t = (ti )ni=1 have distinct nonzero components, n . If we specify (9.1) for a T-Bezoutian BezT (u, v), then we obtain t−1 = t1i i=1

for the oﬀ diagonal entries cij of C = Vn (t)BezT (u, v)Vn (t−1 )T the relation cij = −

u(ti )v(tj ) − v(ti )u(tj ) 1−n tj . ti − t j

For the diagonal entries we obtain using l’Hospital’s rule . cii = (v (ti )u(ti ) − u (ti )v(ti ))t1−n i In the same way as in Proposition 9.1 we derive the following. Proposition 9.5. Let α ∈ C be such that w(t) = u(t) − αv(t) has simple nonzero roots t1 , . . . , tn , and let t = (ti )ni=1 . Then Vn (t)BezT (u, v)Vn (t−1 )T = diag(γi )ni=1 , where γi = (v (ti )u(ti ) − u (ti )v(ti ))t1−n . i Let u be conjugate-symmetric and v conjugate-skewsymmetric. Then the matrix BezT (u, v) is Hermitian. For purely imaginary α, the roots of w(t) = u(t) − αv(t) are located symmetrically with respect to the unit circle T.

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87

Corollary 9.6. Let in Proposition 9.5 the polynomial u be conjugate-symmetric and v(t) be conjugate-skewsymmetric, and let α ∈ iR. Furthermore, let t1 , . . . , tr be the −1 −1 (simple) roots of w(t) on T and tr+1 , tr+2 = tr+1 , . . . , tn−1 , tn = tn−1 the (simple) roots of w(t) outside T. Then

0 0 γ r+1 γ n−1 ,..., . Vn (t)BezT (u, v)Vn (t)∗ = diag γ1 , . . . , γr , γr+1 0 γn−1 0 In particular, sgn BezT (u, v) =

r

sgn γi .

i=1

3. Conﬂuent case. Here we need the following generalization of a Vandermonde m m matrix. Let t = (ti )m i=1 and r = (ri )i=1 ∈ N . We denote by Vn (ti , ri ) the ri × n matrix ⎡ ⎤ 1 ti t2i ... tn−1 i ⎢ 0 1 2ti ⎥ . . . (n − 1)tn−2 i ⎢ ⎥ Vn (ti , ri ) = ⎢ .. ⎥ . . .. .. ⎣ . ⎦ n−1 n−ri 0 ... 0 1 ... ri −1 ti and introduce the matrix

⎤ Vn (t1 , r1 ) ⎥ ⎢ .. Vn (t, r) = ⎣ ⎦ , . Vn (tm , rm ) ⎡

which is called conﬂuent Vandermonde matrix. Now we show that in case that u(t) has multiple roots Bezoutians can be reduced to block diagonal form with the help of conﬂuent Vandermonde matrices Vn (t, r) . We restrict ourselves to the case of H-Bezoutians. The case of TBezoutians is analogous. First we consider the special single node case t = 0, r = r. Suppose that u = (ui )n+1 i=1 and u(t) has the root t = 0 with multiplicity r, i.e., u1 = u2 = · · · = ur = 0. Obviously, Vn (0, r) = Ir O . Hence Vn (0, r)BVn (0, r)T is the r × r leading principal submatrix of B = BezH (u, v). We denote this matrix by Γ(0) and observe, taking Theorem 3.2 into account, that ⎤⎡ ⎤ ⎡ ⎤ ⎡ ur+1 w1 v1 ⎥⎢ ⎥ ⎢ ⎢ .. .. ⎥ , . . Γ(0) = ⎣ ... . . . ⎦⎣ ⎦=⎣ .. .. . . ⎦ vr

...

ur+1

v1

...

u2r+1

where 2n−r+1 i=1

wi ti−1 = u(t)v(t)t−r .

w1

...

wr

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Taylor expansion gives us ui =

1 u(i−1) (0) (i − 1)!

and an analogous expression for vi . Suppose now that t0 is a root of u(t) with multiplicity r. We consider the = [ bij ]n (t) = v(t−t0 ) and B ). (t) = u(t−t0 ) and v u, v polynomials u i,j=1 = BezH ( Then, for k = 0, 1, . . . , (k) (0) = u(k) (t0 ) , u

(k) (0) = v(k) (t0 ) v

and bij

= =

Hence

∂ i+j−2 1 B(t, s) |(0,0) (i − 1)!(j − 1)! ∂ti−1 ∂sj−1 ∂ i+j−2 1 B(t, s) |(t0 ,t0 ) . (i − 1)!(j − 1)! ∂ti−1 ∂sj−1

⎡

v 1 ⎢ .. Γ(t0 ) = ⎣ . v r

⎤⎡ ⎥⎢ ⎦⎣

..

. ...

v 1

u r+1

. .. ...

⎤ ⎡ u r+1 ⎥ ⎢ .. ⎦=⎣ . u 2r+1 w1

1 u(i−1) (t0 ), analogously for v i , and (i − 1)! u(t)v(t)(t − t0 )−r . We arrive at the following. where u i =

. .. ...

2n−r+1

⎤ w1 .. ⎥ , . ⎦ wr

wi (t − t0 )i−1 =

i=1

Proposition 9.7. Let t1 , . . . , tm be the (diﬀerent) roots of u(t) and r1 , . . . , rm the m corresponding multiplicities, t = (ti )m i=1 and r = (ri )i=1 . Then Vn (t, r)BezH (u, v)Vn (t, r)T = diag(Γi )m i=1 , where

⎡

⎢ Γi = ⎣

. .. wi1

...

⎤ wi1 .. ⎥ . ⎦ wiri

and

2n−r i +1

wij (t − ti )j−1 = u(t)v(t)(t − ti )−ri .

j=1

The case v(t) = 1 provides a formula for the inverse of conﬂuent Vandermonde matrices. Corollary 9.8. Let t = (t1 , . . . , tq ) have distinct components, and let u(t) =

q

(t − ti )ri ,

i=1

where n = r1 + · · · + rq . Then the inverse of the conﬂuent Vandermonde matrix Vn (t, r), r = (r, . . . , rq ), is given by m Vn (t, r)−1 = B(u)Vn (t, r)T diag (Γ−1 i )i=1 ,

Introduction to Bezoutians ⎡ ⎢ where Γi = ⎣ wi1

. .. ...

⎤ wi1 .. ⎥ with w = ij . ⎦ wiri

u(j+ri ) (ti ) (j+ri −1)!

89

.

We leave it to the reader to state the analogous properties of T-Bezoutians.

10. Root localization problems In this section we show the importance of Bezoutians, Hankel and Toeplitz matrices for root localization problems. Throughout the section, let F = C. 1. Inertia of polynomials. Let C be a simple oriented closed curve in the extended complex plane C∞ = C ∪ {∞} dividing it into an “interior” part Ω+ and an “exterior” part Ω− . We assume that the domain Ω+ is situated left from C if a point moves along C in positive direction. The inertia of the polynomial u(t) ∈ Cn+1 (t) with respect to C is, by deﬁnition, a triple of nonnegative integers inC (u) = (π+ (u), π− (u), π0 (u)) , where π± (u) is the number of zeros of u(t) in Ω± , respectively, and π0 (u) is the number of zeros on C. In all cases multiplicities are counted. We say that u(t) ∈ Cn+1 (t) has a zero at ∞ with multiplicity r if the r leading coeﬃcients of u(t) are zero. By a root localization problem we mean the problem to ﬁnd the inertia of a given polynomial with respect to a curve C. In the sequel we deal only with the cases that C is the real line R, the imaginary line i R, or the unit circle T. We relate the inertia of polynomials to inertias of Hermitian matrices, namely to Bezoutians. Recall that the inertia of an Hermitian matrix A is the triple In A = (p+ (A), p− (A), p0 (A)) , where p+ (A) is the number of positive, p− (A) the number of negative eigenvalues (counting multiplicities), and p0 (A) the nullity of A. Clearly, the inertia of A is completely determined by the rank and the signature of A. The importance of the relation consists in the fact that the inertia of Bezoutians can be computed via recursive algorithms for triangular factorization, which were described in Section 5. 2. Inertia with respect to the real line. Let u(t) be a given monic polynomial of degree n, q(t) its real and p(t) its imaginary part, i.e., u(t) = q(t) + i p(t). We consider the matrix 1 B = BezH (u, u). (10.1) 2i 1 For example, if u(t) = t − c, then B = 2i (c − c) = Im c. Hence c is in the upper half-plane if and only if B > 0 . In general, we have

1 (t − s)B(t, s) = (q(t) + ip(t))(q(s) − ip(s)) − (q(t) − ip(t))(q(s) + ip(s)) 2i =

p(t)q(s) − q(t)p(s) .

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Hence, B = BezH (p, q) .

(10.2)

In particular, we see that B is a real symmetric matrix. The following is usually referred to as Hermite’s theorem. Theorem 10.1. Let u(t) be a monic polynomial of degree n, inR (u) = (π+ (u), π− (u), π0 (u)), and B be deﬁned by (10.1) or (10.2). Then the signature of B is given by sgn B = π+ (u) − π− (u) . In particular, B is positive deﬁnite if and only if u(t) has all its roots in the upper half-plane. Furthermore, if u(t) and u(t) are coprime, then In B = inR (u). Proof. Let d(t) be the greatest common divisor of u(t) and u(t) and δ its degree. Then d(t) is also the greatest common divisor of p(t) and q(t), and let u(t) = d(t)u0 (t). Then u(t) = d(t)u0 (t), since d(t) is real. According to (2.5) we have

B0 O ∗ Res(d, u0 ) , B = Res (d, u0 ) O O 1 where B0 = 2i BezH (u0 , u0 ). Since d(t) and u0 (t) are coprime, Res(d, u0 ) is nonsingular. By Sylvester’s inertia law we have sgn B = sgn B0 . We ﬁnd now sgn B0 . Let z1 be a root of u0 (t) and u0 (t) = (t−z1 )u1 (t). Then u0 (t) = (t−z1 )u1 (t). Taking into account that

1 (t − z1 )(s − z 1 ) − (t − z 1 )(s − z1 ) = Im z1 2i t−s we obtain using (2.5) B0 = Res (t − z 1 , u1 )

∗

B1 0T

0 Im z1

Res (t − z 1 , u1 ) ,

1 where B1 = 2i BezH (u1 , u1 ) . Repeating these arguments for the other roots zk of u0 (t) (k = 2, . . . , n − δ) we conclude that there is a matrix R such that

B0 = R∗ diag (Im z1 , . . . , Im zn−δ ) R . Thus, B0 is congruent to the diagonal matrix of the Im zi . Applying Sylvester’s inertia law we obtain sgn B = sgn B0 =

n−δ

sgn (Im zi ) = π+ (u) − π− (u) ,

i=1

which proves the main part of the theorem.

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If u(t) and u(t) are coprime, then B is nonsingular, thus p0 (B) = 0, and π0 (u) = 0. Hence π+ (u) + π− (u) = n. Consequently π± (u) = p± (B), and the theorem is proved. If u(t) is a real polynomial, then the Bezoutian B is zero, so all information about the polynomial is lost. It is remarkable that in the other cases information about the polynomial is still contained in B. Example 10.2. Let u(t) = (t − z0 )d(t), where d(t) is real and z0 is in the upper half-plane. Then B = Im z0 dd∗ . This matrix has signature 1 saying that π+ (u) − π− (u) = 1 but not specifying the location of the roots of d(t). Nevertheless, the polynomial d(t) and so information about u(t) can be recovered from B. Recall that according to the results in Section 5 the Euclidian algorithm applied to the polynomials p(t) and q(t) provides a method to compute the signature of B in O(n2 ) operations. We know from Proposition 7.18 that the Bezoutian B = BezH (p, q) and the p Hankel matrix Hn = Hn − q have the same inertia. Hence we can conclude the following. Corollary 10.3. Let p(t) , q(t) ∈ Rn+1 (t) be two coprime polynomials, where q(t) is monic with degree n and u(t) = q(t) + ip(t). Then

p . inR (u) = In Hn − q Theorem 10.1, gives a complete answer to the problem to ﬁnd the inertia of a polynomial only if u(t) has no real and conjugate complex roots. In the other cases we have only partial information. More precisely, if δ denotes the number of real roots of u(t), then 1 π± (u) = p± (B) + (p0 (B) − δ). 2 Note that 12 (p0 (B) − δ) is the number of conjugate complex pairs of roots of u(t). The number δ is also the number of real roots of the greatest common divisor d(t) of u(t) and u(t). Clearly, d(t) is a real polynomial. Thus we are led to the problem to count the number of real roots of a real polynomial. This problem will be discussed next. 3. Real roots of real polynomials. Let p(t) be a real polynomial of degree n. We p (t) (t) consider the rational function p(t) and the Hankel matrix Hn pp(t) . This is just our Example 7.2. According to Proposition 7.17 the rank of this matrix is equal to the number of diﬀerent roots of p(t) and, due to Corollaries 7.11 and 1.2, the signature is the number of diﬀerent real roots of p(t). Let π0 (p) denote the number of diﬀerent real roots of p(t). Taking also into account Proposition 7.18, we have now the following Theorem of Jacobi-Borchardt.

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Theorem 10.4. The number of diﬀerent real roots π0 (p) of the real polynomial p is given by p (t) π0 (p) = sgn BezH (p, p ) = sgn Hn . p(t) Example 10.5. Let p(t) = t4 − 1. ⎡ 1 0 0 ⎢ 0 0 0 p (t) ⎢ H4 = 4⎣ 0 0 1 p(t) 0 1 0

Then ⎤ 0 1 ⎥ ⎥, 0 ⎦ 0

⎡

0 ⎢ 0 ⎢ BezH (p, p ) = 4 ⎣ 1 0

0 1 0 0

1 0 0 0

⎤ 0 0 ⎥ ⎥. 0 ⎦ 1

Both matrices have signature 2, which conﬁrms Theorem 10.4. Theorem 10.4 completely solves the root localization problem for real polynomials in case of simple roots. The multiple roots are just the roots of the greatest common divisor p1 (t) of p(t) and p (t). If we ﬁnd the numbers of diﬀerent real roots of p1 (t) by Theorem 10.4, then we can obtain the number of real roots with multiplicity of at least two. If we continue we obtain the number of diﬀerent real roots with multiplicity of at least 3, and so on. From the algorithmic view point we have to apply the Euclidian algorithm again and again. If it terminates at a certain polynomial d(t), then we continue with d(t) and d (t) until the remainder is a constant. It is remarkable that all together there are not more than n steps. This also concerns the general root localization problem for a complex polynomial u(t) which was discussed above in Section 10.2. First we apply the Euclidian algorithm to the real and imaginary parts of u(t), and when it terminates at a non-constant d(t), then we continue with the Euclidian algorithm for d(t) and d (t). Again we have at most n steps if the degree of the original polynomial is n . 4. Inertia with respect to the imaginary axis. To ﬁnd the inertia of a real polynomial with respect to the imaginary axis iR is a very important task in many applications, because it is related to the question of stability of systems. In order to avoid confusion, let us point out that if iniR (u) = (π+ (u), π− (u), π0 (u)) , then according to the deﬁnition above π+ (u) is the number of roots of u(t) with negative real part and π− (u) those with positive real part. Clearly, the imaginary axis case can easily be transformed into the real line case by a transformation of the variable. It remains to study the speciﬁcs that arises from the fact that the polynomial under investigation is real. Suppose that p(t) is a real polynomial of degree n. We set u(t) = p(it). Then

Furthermore, if p(t) =

n+1 j=1

in iR (p) = in R (u) . pj tj−1 , then u(t) admits a representation u(t) = a(t2 ) + i t b(t2 ) ,

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where, for odd n = 2m + 1 , a(t) = p1 − p3 t + · · · + (−1)m p2m+1 tm ,

b(t) = p2 − p4 t + · · · + (−1)m p2m+2 tm , (10.3)

and for even n = 2m , a(t) = p1 − p3 t + · · · + (−1)m p2m+1 tm ,

b(t) = p2 − p4 t + · · · + (−1)m−1 p2m tm−1 . (10.4) 1 From (10.2) and Proposition 2.8 we conclude that the matrix B = 2i BezH (u, u) is congruent to the direct sum of the matrices B0 = BezH (tb, a)

and B1 = BezH (b, a) .

(10.5)

Using Theorem 10.1 we arrive at the following. Theorem 10.6. Let p(t) be a real polynomial of degree n and let a(t) and b(t) be deﬁned by (10.3) or (10.4), depending on whether n is odd or even, and let B0 and B1 be given by (10.5). Then sgn B0 + sgn B1 = π+ (p) − π− (p), where π+ (p) denotes the number of roots of p(t) in the left and π− (p) the number of roots in the right half-plane. In particular, the roots of p(t) lie entirely in the left half-plane if and only both matrices B0 and B1 are positive deﬁnite. Furthermore, if a(t) and b(t) are coprime, then iniR (p) = In B0 + In B1 . 5. Roots on the imaginary axis and positive real roots of real polynomials. In order to get a full picture about the location of the roots of the real polynomial p(t) with respect to the imaginary axis we have to ﬁnd the number π0 of all roots on the imaginary axis, counting multiplicities. As a ﬁrst step we ﬁnd the number π0 of diﬀerent roots on iR. This number is equal to the number of diﬀerent real roots of the greatest common divisor of a(t2 ) and tb(t2 ), where a(t) and b(t) are deﬁned by (10.3) or (10.4). Let p(0) = 0 and d(t) be the greatest common divisor of a(t) and b(t). Then the greatest common divisor of a(t2 ) and tb(t2 ) equals d(t2 ). The number of real roots of the polynomial d(t2 ) equals twice the number of positive real roots of d(t). Thus, we are led to the problem to count the roots of a real polynomial on the positive real half-axis. Let p(t) be a real polynomial of degree n and p(0) = 0. We consider the (t) function f1 (t) = tp p(t) . This function has a partial fraction decomposition f1 (t) =

n i=1

n t ti =n+ , t − ti t − ti i=1

where t1 , . . . , tn are the roots of p(t). This leads to a Vandermonde factorization similar to the factorization (7.10) in Example 7.9. From this factorization we conclude that sgn Hn (f1 ) = δ+ − δ− ,

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where δ+ denotes the number of diﬀerent positive and δ− the number of diﬀerent negative real roots of p(t). Using Proposition 7.18 and the result of Theorem 10.4, which is sgn BezH (p, p ) = δ+ + δ− , we conclude the following. Theorem 10.7. Let p(t) be as in Theorem 10.6 with p(0) = 0 . The number of positive real roots δ+ of p(t) is given by 1 (sgn BezH (p, p ) + sgn BezH (p, tp )) . 2 Furthermore, all roots of p(t) are real and positive if and only if BezH (p, tp ) is positive deﬁnite. δ+ =

Example 10.8. For p(t) = t4 − 1 we obtain BezH (p, tp ) = 4 J4 . This matrix has a signature equal to zero. Thus (cf. Example 10.5), 1 (sgn BezH (p, p ) + sgn BezH (p, tp )) = 1 2 which conﬁrms Theorem 10.7. 6. Inertia with respect to the unit circle. Now we discuss the problem how to ﬁnd the inertia inT (u) = (π+ (u), π− (u), π0 (u)) of a complex monic polynomial u(t) of degree n with respect to the unit circle T. According to the deﬁnition in Section 10.1, π+ (u) is the number of roots inside the unit circle, π− (u) is the number of roots outside the unit circle, and π0 (u) the number of roots on the unit circle. We consider the matrix B = BezT (u# , u) . (10.6) For example, if n = 1 and u(t) = t − c, then B = 1 − |c|2 . Thus c belongs to the open unit disk if and only if B > 0. A general u(t) can be represented as u(t) = u+ (t) + iu− (t), where u± are conjugate symmetric. We have now that the polynomial (1 − ts)B(t, s) is equal to (u+ (t) − iu− (t))(u+ (s) + iu− (s)) − ((u+ (t) + iu− (t))(u+ (s) − iu− (s))). Thus (1 − ts)B(t, s) = 2i(u+ (t)u− (s) − u− (t)u+ (s)) , which means B = 2i BezT (u+ , u− ) .

(10.7)

The following is usually referred to as the Schur-Cohn theorem. It can be proved with the same arguments as Hermite’s theorem (Theorem 10.1). Theorem 10.9. Let u(t) be a monic polynomial of degree n and B be deﬁned by (10.6) or (10.7). Then the signature of B is given by sgn B = π+ (u) − π− (u) . In particular, B is positive deﬁnite if and only if u(t) has all its roots in the open unit disk. Furthermore, if u(t) and u# (t) are coprime, then In B = inT (u).

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Theorem 10.9 provides full information about the inertia of u(t) only if u(t) has no roots on the unit circle and symmetric with respect to the unit circle or, what is the same if u(t) and u# (t) are coprime. To complete the picture we still have to ﬁnd the inertia of the greatest common divisor of u(t) and u# (t), which is a conjugate-symmetric polynomial. 7. Roots of conjugate-symmetric polynomials. Let w(t) be a monic conjugatesymmetric polynomial of degree n and t1 , . . . , tn its roots. Then w(t) =

n

(t − tk ) = w (t) = #

k=1 −1

which implies tk

n

(1 − tk t) ,

k=1

= tk , k = 1, . . . , n . We consider the function f (t) =

(w )# (t) , w# (t)

where (w )# = Jn w This function has a partial fraction decomposition (cf. Example 8.2) n 1 . f (t) = 1 − tk t k=1

From this representation we can see that the Toeplitz matrix Tn (f ) is Hermitian and its signature is equal to the number of diﬀerent roots of w(t) on T (cf. Corollary 8.7, (8.10), and Corollary 1.2). Theorem 10.10. For a conjugate-symmetric polynomial w(t), BezT ((w )# , w# ) is Hermitian, and its signature is equal to the number of diﬀerent roots of w(t) on the unit circle.

11. Toeplitz-plus-Hankel Bezoutians Some important results for the Toeplitz and Hankel case can be generalized to matrices which are the sum of such structured matrices. In particular, we will show that the inverse of a (nonsingular) matrix which is the sum of a Toeplitz plus a Hankel matrix possesses again a (generalized) Bezoutian structure. To be more precise we deﬁne the following. 1. Deﬁnition. An n × n matrix B is called Toeplitz-plus-Hankel Bezoutian, brieﬂy T+H-Bezoutian, if there are eight polynomials gi (t), fi (t) (i = 1, 2, 3, 4) of Fn+2 (t) such that 4 gi (t)fi (s) i=1 B(t, s) = . (11.1) (t − s)(1 − ts) In analogy to the Hankel or Toeplitz case we use here the notation B = BezT +H ((gi , fi )41 ).

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H-Bezoutians or T-Bezoutians are also T+H-Bezoutians. For example, the ﬂip matrix Jn introduced in (1.2) is an H-Bezoutian, Jn (t, s) can be written as Jn (t, s) =

tn − sn − tn+1 s + tsn+1 tn − sn = , t−s (t − s)(1 − ts)

which shows that Jn is the T+H-Bezoutian (11.1), where g1 = −f2 = en+1 , g2 = f1 = 1, g3 = f4 = en+2 , g4 = −f3 = e2 . The shift matrix (1.8) is a T-Bezoutian and a T+H-Bezoutian, Sn (t, s) =

t2 − tn+1 sn−1 − ts + tn sn t − tn sn−1 = . 1 − ts (t − s)(1 − ts)

For these examples the sum Sn + Jn is also a T+H-Bezoutian, (Sn + Jn )(t, s) =

(tn + t2 ) − tn+1 (s + sn−1 ) + (tn − 1)sn + t(sn+1 − s) . (t − s)(1 − ts)

But for any vectors u, v, g, h ∈ Fn+1 , n > 3, the rank of the matrix with the generating polynomial (1 − ts)(u(t)v(s) − v(t)u(s)) + (t − s)(g(t)hJ (s) − h(t)gJ (s)) is not expected to be less or equal to 4. This means that the sum of a T- and an H-Bezoutian BezH (u, v) + BezT (g, f ) is, in general, not a T+H-Bezoutian. 2. The transformation ∇T +H . The T+H analogue of the transformations ∇H or ∇T (introduced in Section 2.2 and in Section 2.8) is the transformation ∇T +H mapping a matrix A = [ aij ]ni,j=1 of order n onto a matrix of order n + 2 according to n+2 A = [ ai−1,j − ai,j−1 + ai−1,j−2 − ai−2,j−1 ]i,j=1 . Here we put aij = 0 if i ∈ / {1, 2, . . . , n} or j ∈ / {1, 2, . . . , n}. Denoting Wn = Sn +SnT we have ⎡ ⎤ 0 −eT1 A 0 ∇T +H A = ⎣ Ae1 AWn − Wn A Aen ⎦ . (11.2) 0 −eTn A 0 The generating polynomial of ∇T +H A is (∇T +H A)(t, s) = (t − s)(1 − ts)A(t, s) .

(11.3)

Hence a matrix B is a T+H-Bezoutian if and only if rank∇T +H B ≤ 4 . Recall that the n × n matrix in the center of (11.2) is the matrix ∇(A) introduced in (1.11). In other words, the transformation ∇ is a restriction of ∇T+H , and it is clear that T+H-Bezoutians are quasi-T+H matrices, but not vice versa. 3. Uniqueness. Diﬀerent vector systems {gi , fi }4i=1 , {g i, f i }4i=1 may produce the same T+H-Bezoutian.

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= BezT +H ((g i , f i )41 ) if and Note that B = BezT +H ((gi , fi )41 ) is equal to B only if ∇T +H B = ∇T +H B . To answer the questions under which conditions this happens we use the following lemma. Lemma 11.1. Let Gj , Fj (j = 1, 2) be full rank matrices of order m × r, n × r, respectively, r = rank Gj = rank Fj . Then G1 F1T = G2 F2T

(11.4)

if and only if there is a nonsingular r × r matrix ϕ such that G2 = G1 ϕ , F1 = F2 ϕT .

(11.5)

Proof. Assume there is a nonsingular ϕ so that G2 = G1 ϕ and F2T = ϕ−1 F1T , then G1 F1T = G2 F2T . Now let (11.4) be satisﬁed and A = G1 F1T . The image of A is spanned by the columns of G1 as well as of G2 . Thus there exists a nonsingular matrix ϕ so that G2 = G1 ϕ. With the same arguments for AT we obtain that there is a nonsingular matrix ψ so that F2 = F1 ψ. Hence G1 F1T = G2 F2T = G1 ϕψ T F1T .

(11.6)

Since G1 , F1 are of full rank they are one-sided invertible, and we conclude from (11.6) that ϕ · ψ T = Ir . be n × n T+H-Bezoutians and G, G, F, F be full rank matrices with Let B, B = rank F ≤ 4 r = rank G = rank F ≤ 4 , r = rank G allow the following rank decomposisuch that the matrices ∇T+H B and ∇T+H B tions =G F T . ∇T+H B = GF T , ∇T+H B coincide if and only if r = r , Proposition 11.2. The T+H-Bezoutians B and B and there is a nonsingular r × r matrix ϕ so that = Gϕ , F = F ϕT . G To specify this for nonsingular Bezoutians we make the following observation. Proposition 11.3. Let B be an n × n matrix (n ≥ 2) with rank ∇T+H B < 4. Then B is a singular matrix. Proof. Let us prove this by contradiction. Assume B is nonsingular and ∇T+H B < 4. Taking (11.2) into account elementary considerations show that ∇T+H B allows the following decomposition ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎡ ⎤ 0 1 0 0 ∇T+H B = ⎣ Be1 ⎦ [1 ∗ 0 ] +⎣ Ben ⎦ [ 0 ∗ 1] −⎣ ∗ ⎦ [ 0 eT1 B 0 ] −⎣ ∗ ⎦ [ 0 eTnB 0 ], 0 1 0 0 (11.7)

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where ∗ stands for some vector of Fn . Due to the nonsingularity of B its ﬁrst and last rows as well as its ﬁrst and last columns are linearly independent. Thus, ⎤ ⎡ 1 ∗ 0 ⎡ ⎤ 0 0 1 0 ⎥ ⎢ ∗ 1 ⎥ ⎢ 0 ⎢ ⎥ ⎥ = 4, ⎢ rank ⎣ Be1 Ben ∗ ∗ ⎦ = rank ⎢ T ⎥ ⎣ 0 e1 B 0 ⎦ 0 0 0 1 0 eTn B 0 which contradicts rank ∇T+H B < 4.

Corollary 11.4. If rank ∇T+H B < 4 then the ﬁrst and the last rows (or the ﬁrst and the last columns) of B are linearly dependent. For T-(or H-)Bezoutians B , the condition rank ∇T B < 2 (or rank ∇H B < 2) leads to B ≡ 0. But in the T+H case nontrivial T+H-Bezoutians B with rank ∇T+H B < 4 exist. Examples are B = In + Jn (n ≥ 2) and split Bezoutians introduced in Section 2.11. In these cases rank ∇B ≤ 2 . Now we present the result for the nonsingular case. Proposition 11.5. The nonsingular T+H-Bezoutians = BezT+H (( B = BezT+H ((gi , fi )41 ) and B gi , fi )41 ) coincide if and only if there is a nonsingular 4 × 4 matrix ϕ such that 1 g 2 g 3 g 4 ] [ g1 g2 g3 g4 ] ϕ = [ g and

[ f 1 f 2 f 3 f 4 ] ϕT = [ f1 f2 f3 f4 ] .

4. Inverses of T+H-Bezoutians. Recall that in the Hankel and Toeplitz cases we proved that a nonsingular matrix is an H- or a T-Bezoutian if and only if it is the inverse of a Hankel or of a Toeplitz matrix, respectively (see Sections 4.1, 4.4, 7.6, 8.7). Such an assertion is also true in the T+H case. We start with proving the following part of it. Theorem 11.6. Let B be a nonsingular T+H-Bezoutian. Then B −1 is a T+H matrix. Proof. Taking Proposition 11.3 into account we have rank ∇T +H B = 4 , and a rank decomposition of ∇T +H B is of the form (11.7). In particular, this means that there are vectors zi ∈ Fn , i = 1, 2, 3, 4, such that BWn − Wn B = Be1 zT1 + Ben zT2 + z3 eT1 B + z4 eTn B . Applying B −1 from both sides this equality leads to B −1 Wn − Wn B −1 = −(e1 zT1 B −1 + en zT2 B −1 + B −1 z3 eT1 + B −1 z4 eTn ) . Thus, the matrix of order n − 2 in the center of ∇(B −1 ) is the zero matrix. By Proposition 1.4 this proves that B −1 is a T+H matrix.

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In the next section we will show that the converse is also true, i.e., the inverse of a (nonsingular) T+H matrix is a T+H-Bezoutian.

12. Inverses of T+H-matrices We consider now n × n matrices Rn which are the sum of a Toeplitz matrix Tn and a Hankel matrix Hn . For our purposes it is convenient to use a representation (1.5) for m = n, Hn = Tn (b)Jn , b = (bi )n−1 Tn = Tn (a) , a = (ai )n−1 i=1−n , i=1−n , ⎡ ⎤ ⎡ ⎤ a0 b1−n . . . . . . a1−n b0 ⎢ .. ⎥ + ⎢ .. .. ⎥ . (12.1) .. . Rn = Tn (a)+Tn (b)Jn = ⎣ ... . . ⎦ ⎣ . .. . ⎦ a0 . . . bn−1 an−1 . . . b0 We want to prove that the inverse of a T+H matrix Rn is a T+H-Bezoutian and even more, we want to present inversion formulas Rn−1 = BezT +H ((gi , fi )41 ) . Thus, we have to answer the question how to obtain the vectors gi , fi , i = 1, 2, 3, 4. Note that representations of inverses of T+H matrices as T+H-Bezoutians allow fast matrix-vector multiplication by these matrices (in case F = C see [38], in case F = R [40], [42]). 1. Fundamental systems. Besides the nonsingular T+H matrix Rn of (12.1) we consider the (n − 2) × (n + 2) T+H matrices ∂Rn , ∂RnT obtained from Rn , RnT after deleting the ﬁrst and last rows and adding one column to the right and to the left by preserving the T+H structure, ⎡ ⎤ ⎡ ⎤ a1 . . . a2−n a1−n b1 b2 a2 b1−n b2−n . . . ⎢ a3 ⎢ a2 . . . a3−n a2−n ⎥ b2 b3 ⎥ ⎢ ⎥ ⎢ b2−n b3−n . . . ⎥ ∂Rn = ⎢ . + ⎢ ⎥ .. .. .. .. .. .. .. ⎥ , ⎣ .. ⎣ ⎦ . . . . . . . ⎦ ⎡ ⎢ ⎢ ∂RnT = ⎢ ⎣

an−1

an−2

...

a−1

a−2

a−2 a−3 .. .

a−1 a−2 .. .

... ...

an−2 an−3 .. .

an−1 an−2 .. .

a2−n

...

a1

a2

b−2

b−1

...

bn−2

b1−n ⎥ ⎢ b2−n ⎥ ⎢ ⎥+⎢ .. ⎦ ⎣ . b−2

b2−n b3−n .. .

... ...

b1 b2 .. .

b−1

...

bn−2

⎤ ⎡

bn−1 (12.2) ⎤ b2 b3 ⎥ ⎥ .. ⎥ . . ⎦

bn−1 (12.3) Since Rn is nonsingular both matrices ∂Rn and ∂RnT are of full rank, which means a1−n

dim ker ∂Rn = dim ker ∂RnT = 4 . Any system of eight vectors {ui }4i=1 , {vi }4i=1 , where {ui }4i=1 is a basis of ker ∂Rn and {vi }4i=1 is a basis of ker ∂RnT , is called fundamental system for Rn . The reason for this notation is that these vectors completely determine the inverse Rn−1 . In order to understand this we consider ﬁrst a special fundamental system.

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Hereafter we use the following notation. For a given vector a = (aj )n−1 j=1−n we deﬁne a± = (a±j )nj=1 , (12.4) where a±n can be arbitrarily chosen. The matrix ∇(Rn ) = Rn Wn − Wn Rn allows a rank decomposition of the form, ∇(Rn ) = −(a+ + bJ− )eT1 − (aJ− + b+ )eTn + e1 (a− + bJ− )T + en (aJ+ + b+ )T . (12.5) Multiplying (12.5) from both sides by Rn−1 we obtain a rank decomposition of ∇(Rn−1 ). Proposition 12.1. We have ∇ Rn−1 = x1 y1T + x2 y2T − x3 y3T − x4 y4T ,

(12.6)

where xi (i = 1, 2, 3, 4) are the solutions of Rn x1 = a+ + bJ− , Rn x2 = aJ− + b+ , Rn x3 = e1 , Rn x4 = en ,

(12.7)

and yi (i = 1, 2, 3, 4) are the solutions of RnT y1 = e1 , RnT y2 = en , RnT y3 = a− + bJ− , RnT y4 = aJ+ + b+ .

(12.8)

According to (12.2), (12.3) we obtain the following fundamental system for Rn . Proposition 12.2. Let xi , yi ∈ Fn be deﬁned by (12.7), (12.8). The vector system ⎧ ⎤ ⎤ ⎤ ⎤⎫ ⎡ ⎡ ⎡ ⎡ 1 0 0 0 ⎬ ⎨ (12.9) u1 = ⎣ −x1 ⎦ , u2 = ⎣ −x2 ⎦ , u3 = ⎣ x3 ⎦ , u4 = ⎣ x4 ⎦ ⎭ ⎩ 0 1 0 0 is a basis of ker ∂Rn , the vector system ⎧ ⎤ ⎤ ⎤ ⎤⎫ ⎡ ⎡ ⎡ ⎡ 0 0 1 0 ⎬ ⎨ v1 = ⎣ y1 ⎦ , v2 = ⎣ y2 ⎦ , v3 = ⎣ −y3 ⎦ , v4 = ⎣ −y4 ⎦ ⎭ ⎩ 0 0 0 1

(12.10)

is a basis of ker ∂RnT . 2. Inversion of T+H matrices. The special fundamental system of Proposition 12.2 deliver the parameters needed in a Bezoutian formula for Rn−1 . This is the initial point for our further considerations. Theorem 12.3. Let Rn be the nonsingular T+H matrix (12.1) and {ui }4i=1 , {vi }4i=1 be the fundamental system for Rn given by (12.9), (12.7), (12.10), (12.8). Then Rn−1 is the T+H-Bezoutian deﬁned by its generating polynomial as follows, Rn−1 (t, s) =

u3 (t)v3 (s) + u4 (t)v4 (s) − u1 (t)v1 (s) − u2 (t)v2 (s) . (t − s)(1 − ts)

(12.11)

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Proof. Since x3 is the ﬁrst, x4 the last column, y1T is the ﬁrst, y2T the last row of Rn−1 we conclude from (11.2) ⎡ ⎤ 0 −y1T 0 ⎢ ⎥ ∇T +H Rn−1 = ⎣ x3 ∇(Rn−1 ) x4 ⎦ . 0 −y2T 0 Taking (12.6) into account this leads to ∇T +H Rn−1 = [ −u1 − u2 u3 u4 ] [ v1 v2 v3 v4 ]T , where the vectors ui and vi are deﬁned in (12.9), (12.10). The inversion formula follows now from (11.3). In particular, this theorem shows that if we want to use the vectors of any fundamental system in a Bezoutian formula for Rn−1 a “normalization” of them is necessary. For this purpose we introduce the following (n + 2) × 4 matrices F = [ e1 en+2 f1 f2 ] , where

G = [ g1 g2 e1 en+2 ] ,

n+1

n+1

f1 = (a1−i + bi−n )i=0 , f2 = (an−i + bi−1 )i=0 , n+1

n+1

g1 = (ai−1 + bi−n )i=0 , g2 = (ai−n + bi−1 )i=0 , with a±n , b±n arbitrarily chosen. We call a fundamental system {ui }4i=1 , {vi }4i=1 for Rn canonical if F T [ u1 u2 u3 u4 ] = GT [ v1 v2 v3 v4 ] = I4 .

(12.12)

Proposition 12.4. A fundamental system {ui }4i=1 , {vi }4i=1 for Rn is canonical if and only if ui is of the form (12.9), (12.7) and vi is of the form (12.10), (12.8) for i = 1, 2, 3, 4 . Proof. If {ui }4i=1 and {vi }4i=1 are canonical then (12.12) means, in particular, that the ﬁrst component of u1 and v3 as well as the last components of u2 and v4 are one. The ﬁrst and last components of the other vectors are zero. Hence there are vectors xi , yi ∈ Fn such that ui , vi are of the form (12.9), (12.10). Now by (12.12) we have

1 0 T . (12.13) [ I+− f1 I+− f2 ] [ x3 x4 ] = 0 1 n+2 the vector I+− h ∈ Fn is deﬁned by Here, for a given vector h = (hi )n+1 i=0 ∈ F

I+− h = (hi )ni=1 . Since (I+− f1 )T = eT1 Rn , (I+− f2 )T = eTn Rn ⎤ 0 0 and since ⎣ x3 ⎦ , ⎣ x4 ⎦ are in ker ∂Rn equality (12.13) leads to 0 0 ⎡

⎤ ⎡

Rn x3 = e1 , Rn x4 = en .

(12.14)

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⎡ ⎤ ⎤ 1 0 Moreover, ⎣ −x1 ⎦ ∈ ker ∂Rn means that Rn x1 = a+ + bJ− and ⎣ −x2 ⎦ ∈ 0 1 J ker ∂Rn means that Rn x2 = a− + b+ . Similar arguments show that yi , i = 1, 2, 3, 4 , are the solutions of (12.8), and the necessity part of the proof is complete. If {ui }4i=1 , {vi }4i=1 are of the form (12.9), (12.7), and (12.10), (12.8) then, obviously, (12.12) is satisﬁed. vi }4i=1 we deﬁne two 4 × 4 Given an arbitrary fundamental system { ui }4i=1 , { nonsingular matrices ΓF , ΓG , 1 u 2 u 3 u 4 ] = ΓF , FT [u

1 v 2 v 3 v 4 ] = ΓG . GT [ v

We conclude that by 1 u 2 u 3 u 4 ] Γ−1 [ u1 u2 u3 u4 ] = [ u F

(12.15)

1 v 2 v 3 v 4 ] Γ−1 [ v1 v2 v3 v4 ] = [ v G

(12.16)

and a canonical fundamental system {ui }4i=1 , {vi }4i=1 is given. Note that for ﬁxed a±n , b±n the canonical fundamental system is unique. The following becomes clear. ui }4i=1 , { vi }4i=1 Theorem 12.5. Let Rn be the nonsingular T+H matrix (12.1) and { −1 be a fundamental system of Rn . Then the inverse Rn is the T+H-Bezoutian (12.11), where {ui }4i=1 , {vi }4i=1 are given by (12.15), (12.16). Let Rn be given by (12.1). Hereafter we use also a representation of Rn which involves the projections P± = 12 (In ± Jn ) onto Fn± introduced in (1.3) and the vectors n−1 c = (cj )n−1 j=1−n = a + b , d = (dj )j=1−n = a − b ,

namely Rn = Tn (c)P+ + Tn (d)P− .

(12.17)

Instead of the solutions xi of (12.7) and the solutions yi of (12.8) we consider now the solutions of the following equations the right-hand sides of which depend on = aJ − b, c, d and c = aJ + b , d R n w1 =

1 1 (c+ + cJ− ) , Rn w2 = (d+ − dJ− ) , Rn w3 = P+ e1 , Rn w4 = P− e1 2 2 (12.18)

and RnT z1 = P+ e1 , RnT z2 = P− e1 , RnT z3 =

1 1 J ( c+ + cJ− ) , RnT z4 = (d + − d− ) . 2 2 (12.19)

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Here we use the notation (12.4). We introduce the vectors ⎡ ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 1 1 0 ˘ 2 = ⎣ −2w2 ⎦ , u ˘ 3 = ⎣ w3 ⎦ , ˘4 = ⎣ ˘ 1 = ⎣ −2w1 ⎦ , u u u 1 −1 0 ⎡ ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 0 0 1 ˘ 1 = ⎣ z1 ⎦ , ˘ 2 = ⎣ z2 ⎦ , ˘ 3 = ⎣ −2z3 ⎦ , v ˘4 = ⎣ v v v 0 0 1

⎤ 0 w4 ⎦ , 0 ⎤ 1 −2z4 ⎦ . −1 (12.20) Now an inversion formula which involves these vectors follows from formula (12.11). Proposition 12.6. Let Rn be the nonsingular T+H matrix (12.17). Then the inverse Rn−1 is given by Rn−1 (t, s) =

˘ 3 (t)˘ ˘ 4 (t)˘ ˘ 1 (t)˘ ˘ 2 (t)˘ u v3 (s) + u v4 (s) − u v1 (s) − u v2 (s) , (t − s)(1 − ts)

(12.21)

vi }4i=1 are deﬁned in (12.20). where {˘ ui }4i=1 , {˘ Proof. Since ˘1 u ˘2 u ˘3 u ˘ 4 ] = [ u1 u2 u3 u4 ] ϕ [u and ˘2 v ˘3 v ˘ 4 ] = [ v1 v2 v3 v4 ] ϕ−1 , ˘1 v [v where ϕ is the block diagonal matrix

1 1 1 1 , ϕ = diag 1 −1 2 1

1 −1

,

the proposition follows from Proposition 11.5 and (12.11).

3. Inversion of symmetric T+H matrices. It is easy to see that a T+H matrix is symmetric if and only if the Toeplitz part has this property. Let Rn be a nonsingular, symmetric T+H matrix (12.1). Then the solutions of (12.7) and (12.8) coincide, y1 = x3 , y2 = x4 , y3 = x1 , y4 = x2 . Using the inversion formula (12.11) Rn−1 is given by the vectors {ui }4i=1 of (12.9), Rn−1 (t, s) =

u3 (t)u1 (s) − u1 (t)u3 (s) + u4 (t)u2 (s) − u2 (t)u4 (s) . (t − s)(1 − ts)

(12.22)

, and the inversion formula (12.21) can be Since a = aJ we have c = c, d = d simpliﬁed as well, Rn−1 (t, s) =

˘ 3 (t)˘ ˘ 1 (t)˘ ˘ 4 (t)˘ ˘ 2 (t)˘ u u1 (s) − u u3 (s) + u u2 (s) − u u4 (s) . (t − s)(1 − ts)

(12.23)

If we have any basis { ui }4i−1 of ker ∂Rn , it remains to compute ΓF , and {ui }4i=1 is given by (12.15).

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We will not consider the skewsymmetric case, since a skewsymmetric T+H matrix is always a pure Toeplitz matrix. (For the skewsymmetric Toeplitz case see Section 4.7.) 4. Inversion of centrosymmetric T+H matrices. If Rn from (12.1) is centrosymmetric, i.e., RnJ = Rn , then in view of Jn Tn (a)Jn = Tn (aJ ) ,

1 1 1 (a + aJ ) + Tn (b + bJ ) Jn . Rn = (Rn + RnJ ) = Tn 2 2 2 Together with Exercises 15, 16 we conclude the following. Proposition 12.7. Let Rn be an n × n T+H matrix. Then the following assertions are equivalent. 1. Rn is centrosymmetric. 2. In the representation (12.1) (resp. (12.17)) the Toeplitz matrices Tn (a) and Tn (b) (resp. Tn(c) and Tn (d)) are symmetric. 3. In the representation (12.1) (resp. 12.17)) a and b (resp. c and d) are symmetric vectors. Corollary 12.8. A centrosymmetric T+H matrix Rn is also symmetric. Moreover, in the centrosymmetric case the representation (12.17) can be written in the form (12.24) Rn = P+ Tn (c)P+ + P− Tn (d)P− . Now we specify the results for general T+H matrices to centrosymmetric T+H matrices Rn . Since Rn is symmetric we can use the simpliﬁcations of the previous subsection. To begin with we observe that the right-hand sides of the ﬁrst and the third equations of (12.18) are symmetric and of the second and the fourth equations are skewsymmetric if we choose cn = c−n , dn = d−n . Since centrosymmetric matrices map symmetric (skewsymmetric) vectors into symmetric (skewsymmetric) vectors, we conclude that the solutions w1 , w3 of (12.18) ˘ 3 of (12.20) are symmetric, whereas w2 , w4 and ˘ 1, u as well as their extensions u ˘ 2, u ˘ 4 are skewsymmetric vectors. This leads to further simpliﬁcations of the inu version formula (12.23). But before presenting this formula let us introduce a more uniﬁed notation, where the subscript + designates symmetric, − skewsymmetric vectors in the fundamental system, ⎡ ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ ⎤ 0 0 1 1 u+ = ⎣ w3 ⎦ , u− = ⎣ w4 ⎦ , v+ = ⎣ −2w1 ⎦ , v− = ⎣ −2w2 ⎦ . (12.25) 0 0 1 −1 Here wi are the solutions of (12.18) which turn obviously into pure Toeplitz equations, Tn (c)w1 = P+ c+ , Tn (d)w2 = P− d+ , Tn (c)w3 = P+ e1 , Tn (d)w4 = P− e1 . (12.26)

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Note that these equations have unique symmetric or skewsymmetric solutions. Thus, the inversion formula (12.23) can be rewritten as a sum of a split Bezoutian of (+)-type and a split Bezoutian of (−)-type. These special Bezoutians were introduced in Section 2.11. Let us use the notations adopted there. Theorem 12.9. Let Rn be a nonsingular, centrosymmetric T+H matrix given by (12.17) and u± , v± be the vectors of Fn+2 deﬁned in (12.25), where the wi are the ± unique symmetric or skewsymmetric solutions of (12.26). Then Rn−1 = B+ + B− , where B± are the split Bezoutians of (±)-type B± = Bezsplit (v± , u± ) . Similar ideas as those of Section 4.5 lead to a slight modiﬁcation of the last theorem. We extend the nonsingular centrosymmetric T+H matrix Rn given by (12.17) to a nonsingular centrosymmetric T+H matrix Rn+2 , such that Rn is its central submatrix of order n. Rn+2 = Tn+2 (c)P+ + Tn+2 (d)P− .

(12.27)

Here c and d are extensions of the original vectors c and d by corresponding ± components c−n = cn , d−n = dn , c−n−1 = cn+1 , d−n−1 = dn+1 . Let x± n+2 , xn be the unique symmetric or skewsymmetric solutions of Tn+2 (c)x+ n+2 = P+ e1 , Tn+2 (d)x− n+2 = P− e1 ,

Tn (c)x+ n = P+ e1 , Tn (d)x− n = P− e1 .

(12.28)

± − (Note that x+ n = w3 , xn = w4 . The solutions xn+2 are up to a constant factor equal to the vectors v± .)

Corollary 12.10. Let Rn+2 be a nonsingular, centrosymmetric extension (12.27) of Rn . Then the equations (12.28) have unique symmetric or skewsymmetric solutions and 1 1 Rn−1 = Bezsplit (x+ Bezsplit (x− n+2 , u+ ) + n+2 , u− ) , r+ r− ⎡ ⎤ 0 ⎣ x± ⎦. where r± are the ﬁrst components of x± n n+2 and u± = 0 If Tn (c) and Tn (d) are nonsingular then Rn is nonsingular. Indeed, taking (12.24) into account Rn u = 0 leads to P+ Tn (c)P+ u = −P− Tn (d)P− u . Hence P+ u = 0 and P− u = 0 which means u = 0. The converse is not true. Take, for example, c = (1, 1, 1) and d = (−1, 1, −1), then T2 (c) and T2 (d) are singular, whereas R2 = 2I2 is nonsingular. One might conjecture that for a nonsingular Rn

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there is always a representation (12.17) with nonsingular Tn (c) and Tn (d). For n = 2 this is true. But this fails to be true for n = 3. Consider, for example, c = (1, 0, 1, 0, 1) and d = (0, 0, 1, 0, 0) . ⎤ ⎡ 3 0 1 1⎣ 0 2 0 ⎦ R3 = T3 (c)P+ + T3 (d)P− = 2 1 0 3 is nonsingular. But T3 (c) is a chess-board matrix (1.4) with c = 1, b = 0 which is singular and uniquely determined in the representation of R3 (cf. Exercise 16). Let us consider besides Rn = Tn (a) + Tn (b)Jn the matrix Rn− = T (a) − T (b)Jn . If Rn is represented in the form (12.24) then the corresponding representation of Rn− is Rn− = P+ Tn (d)P+ + P− Tn(c)P− , which means that the roles of c and d are interchanged. We conclude the following Then

Proposition 12.11. The (symmetric) Toeplitz matrices Tn (c) and Tn (d) are nonsingular if and only if both Rn and Rn− are nonsingular. Proof. We have already shown that the nonsingularity of Tn (c) and Tn (d) implies the nonsingularity of Rn . The nonsingularity of Rn− follows with the same arguments. It remains to show that the singularity of Tn (c) (or Tn (d)) leads to the singularity of Rn or Rn− . Let u be a nontrivial vector such that Tn (c)u = 0 . We split u into its symmetric and skewsymmetric parts u = u+ + u−

(u± ∈ Fn± ).

Clearly, at least one of the vectors u+ or u− is nonzero, and Tn (c)u+ = Tn (c)u− = 0 . Since Rn u+ = Tn (c)u+ , Rn− u− = Tn (c)u− we obtain that Rn or Rn− is singular. This is also obtained if we assume that Tn (d) is singular. 5. Inversion of centro-skewsymmetric T+H matrices. In this subsection let us consider T+H matrices Rn which are centro-skewsymmetric, Rn = −RnJ . Since for an n × n centro-skewsymmetric matrix A, det A = (−1)n det A, all centroskewsymmetric matrices of odd order are singular. Hence we consider here mainly matrices of even order. The centro-skewsymmetric counterpart of Proposition 12.7 is as follows. Proposition 12.12. Let Rn be an n × n T+H matrix. Then the following assertions are equivalent. 1. Rn is centro-skewsymmetric. 2. There is a representation (12.1) (resp. (12.17)) such that the Toeplitz matrices Tn (a) and Tn (b) (resp. Tn (c) and Tn (d)) are skewsymmetric. 3. There is a representation (12.1) (resp. (12.17)) such that a and b (resp. c and d) are skewsymmetric vectors.

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In the remaining part of this subsection we only use such representations. In this case (12.17) can be rewritten as Rn = P− Tn (c)P+ + P+ Tn (d)P− . Its transposed matrix is given by RnT = −(P− Tn (d)P+ + P+ Tn (c)P− ) . = −c. In the equations (12.19) we have c = −d and d In general, Rn is neither symmetric nor skewsymmetric, thus a connection between the solutions of (12.18) and (12.19) is not obvious. If we choose cn = −c−n and dn = −d−n than c− = −c+ , d− = −d+ . Hence the right-hand sides of the equations (12.18), (12.19) are either symmetric or skewsymmetric. Since Rn as a centro-skewsymmetric matrix maps Fn± to Fn∓ , we obtain that the solutions are also either symmetric or skewsymmetric. Let us indicate these symmetry properties again by denoting w+ = w1 , w− = w2 , x− = w3 , x+ = w4 , − = z1 , x + = z2 , w + = z3 , w − = z4 . x ˘j , v ˘ j of (12.20) we set Since these symmetries pass to the augmented vectors u ˘ 1 , v− = u ˘ 2 , u− = u ˘ 3 , u+ = u ˘4 , v+ = u + = v ˘3 , v − = v ˘4 , u − = v ˘1 , u + = v ˘2 . v

(12.29)

The equations (12.18), (12.19) turn into Toeplitz equations, Tn (c)x+ = P− e1 , Tn (c)w+ = P− c+ , Tn(d)x− = P+ e1 , Tn (d)w− = P+ d+ (12.30) and − = P+ c+ , Tn (d) + = P− d+ . x− = −P+ e1 , Tn (c)w x+ = −P− e1 , Tn (d)w Tn (c) (12.31) According to Proposition 12.6 and (11.3) Rn−1 given by the augmented vectors (12.29) of these solutions via T T + T− − v− u T+ + u+ v − − v+ u . ∇T +H Rn−1 = u− v

(12.32)

Now we show how the solutions of (12.30) and (12.31) are related. First we comx− = P+ e1 and Tn (c)x+ = P− e1 for any c ∈ F2n−1 . The pare the equations Tn (c) − following lemma shows that there is an essential diﬀerence between the centrosymmetric and centro-skewsymmetric cases. Lemma 12.13. Let Tn (c) be skewsymmetric. If the equation Tn (c) x− = −P+ e1 is solvable, then equation Tn (c)x+ = P− e1 is also solvable, and if n is even, then the − is a skewsymmetric solution of the ﬁrst equation, then converse is also true. If x a solution of the second equation is given by x+ (t) =

1+t − (t). x 1−t

(12.33)

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x− = −P+ e1 is solvable, then there exists a skewsymmetric solution Proof. If Tn (c) − . Since x − is skewsymmetric, we have x − (1) = 0. Hence (12.33) deﬁnes a x polynomial x+ (t). Moreover, the coeﬃcient vector x+ is symmetric. 1 − (t) and z1 (t) = tz(t). If now Tn (c)z = x Let z ∈ Fn be deﬁned by z(t) = t−1 n 1 n (rk )k=1 , then Tn (c)z = (rk−1 )k=1 , where r0 is some number. In view of Tn (c)(z1 − z) = −P+ e1 , we have 1 1 r0 − r1 = − , r1 = r2 = · · · = rn−1 , rn−1 − rn = − . 2 2 Since the (n − 1) × (n − 1) principal submatrix Tn−1 of Tn (c) is skewsymmetric and the vector z ∈ Fn−1 obtained from z by deleting the last (zero) component is symmetric, the vector Tn−1 z = (rk )n−1 k=1 is skewsymmetric. Hence 1 1 r0 = − , r1 = r2 = · · · = rn−1 = 0, rn = . 2 2 1 We conclude that Tn (c)(z + z ) = −P− e1 . This means that x+ = z + z1 . The proof of the converse direction follows the same lines. One has to take into account that if n is even and x+ is symmetric, then x+ (−1) = 0. Hence 1 z(t) = t+1 x+ (t) is a polynomial. Note that the converse direction of Lemma 12.13 is not true if n is odd. If, for example, ⎤ ⎡ 0 −1 0 0 −1 ⎦ T3 (c) = ⎣ 1 0 1 0 then T3 (c)x+ = P− e1 is solvable but Tn (c) x− = P+ e1 is not. − extends to the augmented The relation between the solutions x+ and x − . We have vectors u+ and u u+ (t) =

1+t − (t). u 1−t

+ (t) = u

1+t u− (t). 1−t

Replacing c by d we obtain

− = P+ c+ . More Now we compare the equations Tn (c)w+ = P− c+ and Tn (c)w − . precisely, we compare the augmented vectors v+ and v − = P+ c+ is Lemma 12.14. Let Tn (c) be skewsymmetric. If the equation Tn (c)w solvable, then the equation Tn (c)w+ = P− c+ is also solvable, and the augmented vectors of these solutions are related via v+ (t) =

1+t − (t). v 1−t

(12.34)

If n is even, then the solvability of Tn (c)w+ = P− c+ implies the solvability of − = P+ c+ . Tn (c)w

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n−1 n+1 Proof. Let T denote the n × (n + 2) matrix T = [ci−j+1 ]i=0 j=0 with c−n = cn . − = P+ c+ , then T v − = 0. Furthermore, if w − is skewsymmetric, then If Tn (c)w 1 − is skewsymmetric. Hence v − (1) = 0 and z(t) = 1−t − (t) is a polynomial. v v We consider the coeﬃcient vector z of z(t) as a vector in Fn+2 and denote the coeﬃcient vector of tz(t) by z1 . Suppose that T z = (rk )n1 , then T z1 = (rk−1 )n1 , where r0 is some number. − and T v − = 0, we conclude that r0 = · · · = rn . Since z − z1 = v n and z ∈ Fn+1 the Let Tn+1 (c) denote the (n + 1) × (n + 1) matrix [ci−j ]i,j=0 vector obtained from z deleting the last (zero) component. Then Tn+1 (c)z = (rk )nk=0 . Here Tn+1 (c) is skewsymmetric and z is symmetric, thus the vector (rk )nk=0 is skewsymmetric. Since all components are equal, it must be the zero vector. We obtain T (z + z1 ) = 0. Observe that z + z1 is symmetric and that T its ﬁrst component is equal to 1. Therefore, z + z1 = v+ = [ 1 − 2w+ 1 ]T for n some symmetric vector w+ ∈ F . This vector is now a solution of the equation Tn (c)w+ = P− c+ . The converse direction is proved analogously taking into account that if n is even, then the length of v+ , which is n + 2, is even. Hence v+ (−1) = 0 and 1 z(t) = 1+t v+ (t) is well deﬁned.

Replacing c by d we obtain + (t) = v

1+t v− (t). 1−t

Taking (12.32), Lemma 12.13 and Lemma 12.14 together we arrive at ∇T +H (Rn−1 )(t, s)

=

u− (t)

1+s 1+s v− (s) − v− (t) u− (s) 1−s 1−s (12.35)

1−s 1−s u+ (s) + u+ (t) v+ (s) , −v+ (t) 1+s 1+s which ﬁnally leads to the following theorem taking (11.3) into account. Theorem 12.15. Let the centro-skewsymmetric T+H matrix Rn be nonsingular and given by (12.17). Then the equations (12.30) are solvable and the generating function of the inverse matrix is given by the augmented vectors of the solutions of these equations via Rn−1 (t, s) = B+ (t, s)

s−1 s+1 + B− (t, s) s+1 s−1

and B± = Bezsplit (u± , v± ) . Note that for a nonsingular matrix Rn all equations (12.30) and (12.31) − is a solution of are uniquely solvable. Moreover, we observe that x = x+ − x − is a solution of Tn (c)w = cJ− . Taking Proposition Tn (c)x = e1 and w = w+ − w

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4.6 into account we obtain the nonsingularity of Tn (c). Analogously, Tn (d) is nonsingular. This leads to the following surprising conclusion. Corollary 12.16. For a centro-skewsymmetric T+H matrix Rn = T (a) + T (b)Jn = T (c)P+ + T (d)P− with skewsymmetric vectors a, b, c, d , the following assertions are equivalent: 1. Rn is nonsingular. 2. Rn− = T (a) − T (b)Jn is nonsingular. 3. T (c) and T (d) are nonsingular. To present the counterpart of Corollary 12.10 let us extend the nonsingular centro-skewsymmetric T+H matrix Rn given by (12.17) to a nonsingular centroskewsymmetric T+H matrix Rn+2 , such that Rn is its central submatrix of order n. Rn+2 = Tn+2 (c)P+ + Tn+2 (d)P− ,

(12.36)

where c and d are extensions of the original vectors c and d by corresponding components c−n = −cn , d−n = −dn , c−n−1 = −cn+1 , d−n−1 = −dn+1 . Let ± x± n+2 , xn be the unique symmetric or skewsymmetric solutions of Tn+2 (c)x− Tn (c)x− n = P+ e1 , n+2 = P+ e1 , + Tn+2 (d)xn+2 = P− e1 , Tn (d)x+ n = P− e1 .

(12.37)

x± , (solutions of (12.31)), thus −u± are the augmented vectors Note that x± n = − ± deﬁned by u± (t) = tx± n (t) . The solutions xn+2 are up to a constant factor equal to the vectors v± . Corollary 12.17. Let Rn+2 be a nonsingular and centro-skewsymmetric extension (12.36) of Rn . Then the equations (12.37) have unique symmetric or skewsymmetric solutions and Rn−1 =

1 1 s−1 s+1 + , Bezsplit (x+ Bezsplit (x− n+2 , u+ ) n+2 , u− ) r+ s + 1 r− s−1

where r± are the ﬁrst components of x± n+2 .

13. Exercises 1. An n × n matrix B is called quasi-T-Bezoutian if ∇T B introduced in (2.17) has rank 2 at most. (a) Show that B is a quasi-T-Bezoutian if and only if BJn is a quasi-HBezoutian (introduced in Section 2.4). (b) State and prove a proposition about the representation of a quasi-TBezoutian that is analogous to Proposition 2.4.

Introduction to Bezoutians 2. The special Toeplitz matrix ⎡ a0 αan−1 ⎢ ⎢ a1 a0 Znα (a) = ⎢ ⎢ . . .. ⎣ .. an−1 ...

... .. . .. . a1

αa1 .. . αan−1 a0

111

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(α ∈ F)

is called α-circulant. (a) Show that the T-Bezoutian of a polynomial u(t) ∈ Fn+1 and tn − α is an α-circulant and each α-circulant is of this form. (b) Show that a T-Bezoutian is a Toeplitz matrix if and only if it is an α-circulant for some α or an upper triangular Toeplitz matrix. 3. Let u(t) be a polynomial of degree n and v(t) a polynomial of degree ≤ n. Describe the nullspace of the transpose of Resp (u, v) (introduced in Section 3.1) in terms of the greatest common divisor of u(t) and v(t). Use this to show that the nullity of Resp (u, v) is, independently of p, equal to the degree of the greatest common divisor of u(t) and v(t). T 4. (a) Show that an n × n matrix A is Toeplitz if and only if I+− ∇T (A)I+− is the zero matrix, where I+− is introduced in (12.14). (b) Show that the product of two nonzero Toeplitz matrices is Toeplitz again if and only if both factors are α-circulants for the same α or both are upper (or lower) triangular Toeplitz matrices. 5. Prove that the nonsingularity of a Toeplitz matrix Tn = [ ai−j ]ni,j=1 follows from the solvability of the equations T n y = e1

and Tn z = (a−n+j−1 )nj=1 ,

where a−n is arbitrarily chosen. Construct a fundamental system from these solutions. 6. Design a Levinson-type algorithm for the solution of the Bezout equation (7.14), i.e., an algorithm that does not rely on successive polynomial division. Compare the complexity of this algorithm with the complexity of the algorithm described in Section 7.7. Hint. Consider ﬁrst the “regular” case in which the degrees of all quotients qi (t) are equal to 1. 7. Let p(t) = p1 + p2 t + p3 t2 + t3 be a monic real polynomial. Show the following theorem of Vyshnegradsky. The polynomial p(t) has all its roots in the left half-plane if and only if all coeﬃcients are positive and p2 p3 > p1 . 8. The factorizations presented in this paper can be used to derive formulas for the determinants of Bezoutians, Hankel, Toeplitz and resultant matrices. To solve the following problems one can use Vandermonde factorization or reduction and take into account that (ti − tj ) , det Vn (t) = i>j

112

G. Heinig and K. Rost where t = (t1 , . . . , tn ) or apply Barnett’s formula and det p(C(u)) =

n

p(ti )

i=1 n

if u(t) = i=1 (t − ti). Suppose that v(t) = ti ) are complex polynomials and m ≤ n. (a) Show that det Hn

v u

= (−1)

n(n−1) 2

m i=1 (t − si )

m n

and u(t) =

n i=1 (t −

(ti − sj ) ,

i=1 j=1

and ﬁnd an analogous formula for det Tn uv . (b) Derive from (a) formulas for the determinants of BezH (u, v), BezT (u, v), and Res (u, v). 9. Find the Toeplitz matrices

m 1 1 + ct and Tn Tn (1 − ct)m 1 − ct 10. Let u(t) and v(t) = v1 (t)v2 (t) be polynomials of degree n. (a) Show that BezH (u, v) = BezH (u, v1 )B(u)−1 BezH (u, v2 ) (b) If u(t) and v(t) are monic, show that BezH (u, v) = B(u)Jn B(v)(C(u)n − C(v)n ) . 11. Let u(t) and v(t) be complex polynomials of degree n and m, respectively, where m ≤ n, and let t1 , . . . , tr be the diﬀerent roots of the greatest common divisor of u(t) and v(t) and ν1 , . . . , νr their multiplicities. Let vectors k (c, ν) be deﬁned by

k i − 1 i−k c . k (c, ν) = ν −1 i=1 Show that the vectors n (ti , k), where k = 1, . . . , νi , i = 1, . . . , r form a basis of the nullspace of BezH (u, v) and the corresponding vectors m+n+p (ti , k) a basis of the nullspace of Resp (u, v) introduced in Section 3.1. 12. Let u(t) and p(t) be coprime polynomials and deg p(t) ≤ deg u(t) = n. Show that for k > n the coeﬃcient vectors of tj u(t), j = 1, . . . , k − n form a basis p of the nullspace of Hk u . 13. Let u(t) be a polynomial with real coeﬃcients of degree n. (a) Describe the number of diﬀerent real roots positive in terms of the tu (t) u signatures of the matrices Hn u(t) and Hn u . (b) Let a and b be real numbers, a < b. Describe the number of diﬀerent real roots of u(t) in the interval [ a, b ] in terms of the signatures of the

Introduction to Bezoutians matrices Hn (g) and Hn g(t) =

u u

113

, where

(t − a)(b − t)u (t) + nt2 u(t) . u(t)

(c) Prove a representation of Res(u, v) which is analogous to that one of Proposition 3.3 but involves BezT (u, v) instead of BezH (u, v) . T 14. Prove that a matrix A is a T+H matrix if and only if the matrix I+− ∇ (A)I+− is the zero matrix, where ∇ (A) is introduced in (1.11) and I+− in (12.14). 15. Let e and eσ denote the vectors of F2n−1 e = ( 1, 1 . . . , 1 ) and eσ = ( (−1)i )2n−1 i=1 . Show that a T+H matrix Rn = Tn (a) + Tn (b)Jn is equal to Rn = Tn (a ) + Tn (b )Jn if and only if, for some α, β ∈ F, a = a + αe + βeσ and b = b − αe − β(−1)n−1 eσ . 16. Let Rn be an n × n T+H matrix given by (12.17) and by Rn = Tn (c )P+ + Tn (d )P− . Show that (a) If n is odd, then c = c, i.e., c is unique, and d is of the form d = d + αe + βeσ for α, β ∈ F. (b) If n is even, then c is of the form c = c + αeσ and d of the form d = d + βe for α, β ∈ F. Here e and eσ are as above. 17. Let Rn be an n × n nonsingular, centro-skewsymmetric T+H matrix given by (12.17). Show that Rn−1 = Tn (c)−1 P− + Tn (d)−1 P+ .

14. Notes 1. The Bezoutian and the resultant matrix have a long history, which goes back to the 18th century. Both concepts grew up from the work of Euler [5] in connection with the elimination of variables for the solution of systems of nonlinear algebraic equations. In 1764, Bezout generalized a result of Euler [1]. In his solution the determinant of a matrix occurred which was only in 1857 shown by Cayley [3] to be the same as that being today called the (Hankel) Bezoutian. For more detailed information see [66]. 2. The classical studies of Jacobi [51] and Sylvester [65] utilized Bezoutians in the theory of separation of polynomial roots. Hermite [49] studied the problem of counting the roots of a polynomial in the upper half-plane. Clearly, this is equivalent to ﬁnding the number of roots in the left half-plane, which is important for stability considerations. A nice review of classical results concerning root localization problems is given in the survey paper [52]; see also [50], [8], [63], [61], [60]. 3. The importance of Bezoutians for the inversion of Hankel and Toeplitz matrices became clear much later. Only in 1974, Lander [55] established the

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fundamental result that the inverse of a (nonsingular) Hankel matrix can be represented as a Bezoutian of two polynomials and that, conversely, any nonsingular Bezoutian is the inverse of a Hankel matrix. Similar results are true for Toeplitz matrices. In [55] also a Vandermonde factorization of Bezoutians was presented. 4. There is a huge number of papers and books dedicated to Bezoutians, resultant matrices and connected problems. Let me recommend some books and survey papers (see also the references therein) to light the younger history and recent developments, to pursue and to accentuate the topic in diﬀerent directions. (This list is far away from being complete!) Books: Gohberg, Lancaster, and Rodman [10], Heinig and Rost [33], Lancaster and Tismenetsky [54], Bini and Pan [2], Fuhrmann [6], Pan [62], Lascoux [56]. Papers: Gohberg, Kaashoek, Lerer, and Rodman [9], Lerer and Tismenetsky [58], Lerer and Rodman [57], Fuhrmann and Datta [7], Gohberg and Shalom [11], Emiris and Mourrain [4], Mourrain and Pan [59].

References [1] E. Bezout. Recherches sur le degr´e des ´equations r´esultants de l’´evanouissement des inconnues, et sur les moyens qu’il convenient d’employer pour trouver ces equations. Mem. Acad. Roy. Sci. Paris, pages 288–338, 1764. [2] D. Bini and V.Y. Pan. Polynomial and matrix computations. Vol. 1. Progress in Theoretical Computer Science. Birkh¨ auser Boston Inc., Boston, MA, 1994. Fundamental algorithms. [3] A. Cayley. Note sur la m´ethode d’´elimination de Bezout. J. Reine Angew. Math., 53:366–367, 1857. [4] I.Z. Emiris and B. Mourrain. Matrices in elimination theory. J. Symbolic Comput., 28(1-2):3–44, 1999. Polynomial elimination – algorithms and applications. [5] L. Euler. Introductio in analysin inﬁnitorum. Tomus primus. Sociedad Andaluza de Educaci´ on Matem´ atica “Thales”, Seville, 2000. Reprint of the 1748 original. [6] P.A. Fuhrmann. A polynomial approach to linear algebra. Universitext. SpringerVerlag, New York, 1996. [7] P.A. Fuhrmann and B.N. Datta. On Bezoutians, Vandermonde matrices, and the Li´enard-Chipart stability criterion. In Proceedings of the Fourth Haifa Matrix Theory Conference (Haifa, 1988), volume 120, pages 23–37, 1989. [8] F.R. Gantmacher. Matrizentheorie. Hochschulb¨ ucher f¨ ur Mathematik [University Books for Mathematics], 86. VEB Deutscher Verlag der Wissenschaften, Berlin, 1986. ˇ With a foreword by D.P. Zelobenko, Translated from the Russian by Helmut Boseck, Dietmar Soyka and Klaus Stengert. [9] I. Gohberg, M.A. Kaashoek, L. Lerer, and L. Rodman. Common multiples and common divisors of matrix polynomials. II. Vandermonde and resultant matrices. Linear and Multilinear Algebra, 12(3):159–203, 1982/83.

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[10] I. Gohberg, P. Lancaster, and L. Rodman. Matrix polynomials. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982. Computer Science and Applied Mathematics. [11] I. Gohberg and T. Shalom. On Bezoutians of nonsquare matrix polynomials and inversion of matrices with nonsquare blocks. Linear Algebra Appl., 137/138:249–323, 1990. [12] I.C. Gohberg and G. Heinig. The inversion of ﬁnite Toeplitz matrices. Mat. Issled., 8(3(29)):151–156, 183, 1973. [13] I.C. Gohberg and G. Heinig. Inversion of ﬁnite Toeplitz matrices consisting of elements of a noncommutative algebra. Rev. Roumaine Math. Pures Appl., 19:623–663, 1974. [14] I.C. Gohberg and G. Heinig. The resultant matrix and its generalizations. I. The resultant operator for matrix polynomials. Acta Sci. Math. (Szeged), 37:41–61, 1975. [15] I.C. Gohberg and G. Heinig. The resultant matrix and its generalizations. II. The continual analogue of the resultant operator. Acta Math. Acad. Sci. Hungar., 28(34):189–209, 1976. [16] G. Heinig. The notion of Bezoutian and of resultant for operator pencils. Funkcional. Anal. i Priloˇzen., 11(3):94–95, 1977. ¨ [17] G. Heinig. Uber Block-Hankelmatrizen und den Begriﬀ der Resultante f¨ ur Matrixpolynome. Wiss. Z. Techn. Hochsch. Karl-Marx-Stadt, 19(4):513–519, 1977. [18] G. Heinig. Verallgemeinerte Resultantenbegriﬀe bei beliebigen Matrixb¨ uscheln. I. Einseitiger Resultantenoperator. Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt, 20(6): 693–700, 1978. [19] G. Heinig. Verallgemeinerte Resultantenbegriﬀe bei beliebigen Matrixb¨ uscheln. II. Gemischter Resultantenoperator. Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt, 20(6):701–703, 1978. [20] G. Heinig. Bezoutiante, Resultante und Spektralverteilungsprobleme f¨ ur Operatorpolynome. Math. Nachr., 91:23–43, 1979. [21] G. Heinig. Generalized resultant operators and classiﬁcation of linear operator pencils up to strong equivalence. In Functions, series, operators, Vol. I, II (Budapest, 1980), volume 35 of Colloq. Math. Soc. J´ anos Bolyai, pages 611–620. North-Holland, Amsterdam, 1983. [22] G. Heinig. Chebyshev-Hankel matrices and the splitting approach for centrosymmetric Toeplitz-plus-Hankel matrices. Linear Algebra Appl., 327(1-3):181–196, 2001. [23] G. Heinig and F. Hellinger. On the Bezoutian structure of the Moore-Penrose inverses of Hankel matrices. SIAM J. Matrix Anal. Appl., 14(3):629–645, 1993. [24] G. Heinig and F. Hellinger. Displacement structure of generalized inverse matrices. Linear Algebra Appl., 211:67–83, 1994. Generalized inverses (1993). [25] G. Heinig and F. Hellinger. Moore-Penrose inversion of square Toeplitz matrices. SIAM J. Matrix Anal. Appl., 15(2):418–450, 1994. [26] G. Heinig and U. Jungnickel. Zur L¨ osung von Matrixgleichungen der Form AX − XB = C. Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt, 23(4):387–393, 1981. [27] G. Heinig and U. Jungnickel. On the Routh-Hurwitz and Schur-Cohn problems for matrix polynomials and generalized Bezoutians. Math. Nachr., 116:185–196, 1984.

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[28] G. Heinig and U. Jungnickel. On the Bezoutian and root localization for polynomials. Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt, 27(1):62–65, 1985. [29] G. Heinig and U. Jungnickel. Hankel matrices generated by Markov parameters, Hankel matrix extension, partial realization, and Pad´e-approximation. In Operator theory and systems (Amsterdam, 1985), volume 19 of Oper. Theory Adv. Appl., pages 231–253. Birkh¨ auser, Basel, 1986. [30] G. Heinig and U. Jungnickel. Hankel matrices generated by the Markov parameters of rational functions. Linear Algebra Appl., 76:121–135, 1986. [31] G. Heinig and U. Jungnickel. Lyapunov equations for companion matrices. Linear Algebra Appl., 76:137–147, 1986. [32] G. Heinig and K. Rost. Algebraic methods for Toeplitz-like matrices and operators, volume 19 of Mathematical Research. Akademie-Verlag, Berlin, 1984. [33] G. Heinig and K. Rost. Algebraic methods for Toeplitz-like matrices and operators, volume 13 of Operator Theory: Advances and Applications. Birkh¨ auser Verlag, Basel, 1984. [34] G. Heinig and K. Rost. On the inverses of Toeplitz-plus-Hankel matrices. Linear Algebra Appl., 106:39–52, 1988. [35] G. Heinig and K. Rost. Inversion of matrices with displacement structure. Integral Equations Operator Theory, 12(6):813–834, 1989. [36] G. Heinig and K. Rost. Matrices with displacement structure, generalized Bezoutians, and Moebius transformations. In The Gohberg anniversary collection, Vol. I (Calgary, AB, 1988), volume 40 of Oper. Theory Adv. Appl., pages 203–230. Birkh¨ auser, Basel, 1989. [37] G. Heinig and K. Rost. Matrix representations of Toeplitz-plus-Hankel matrix inverses. Linear Algebra Appl., 113:65–78, 1989. [38] G. Heinig and K. Rost. DFT representations of Toeplitz-plus-Hankel Bezoutians with application to fast matrix-vector multiplication. Linear Algebra Appl., 284(13):157–175, 1998. ILAS Symposium on Fast Algorithms for Control, Signals and Image Processing (Winnipeg, MB, 1997). [39] G. Heinig and K. Rost. Representations of Toeplitz-plus-Hankel matrices using trigonometric transformations with application to fast matrix-vector multiplication. In Proceedings of the Sixth Conference of the International Linear Algebra Society (Chemnitz, 1996), volume 275/276, pages 225–248, 1998. [40] G. Heinig and K. Rost. Hartley transform representations of inverses of real Toeplitzplus-Hankel matrices. In Proceedings of the International Conference on Fourier Analysis and Applications (Kuwait, 1998), volume 21, pages 175–189, 2000. [41] G. Heinig and K. Rost. Hartley transform representations of symmetric Toeplitz matrix inverses with application to fast matrix-vector multiplication. SIAM J. Matrix Anal. Appl., 22(1):86–105 (electronic), 2000. [42] G. Heinig and K. Rost. Representations of inverses of real Toeplitz-plus-Hankel matrices using trigonometric transformations. In Large-scale scientiﬁc computations of engineering and environmental problems, II (Sozopol, 1999), volume 73 of Notes Numer. Fluid Mech., pages 80–86. Vieweg, Braunschweig, 2000. [43] G. Heinig and K. Rost. Eﬃcient inversion formulas for Toeplitz-plus-Hankel matrices using trigonometric transformations. In Structured matrices in mathematics,

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computer science, and engineering, II (Boulder, CO, 1999), volume 281 of Contemp. Math., pages 247–264. Amer. Math. Soc., Providence, RI, 2001. [44] G. Heinig and K. Rost. Centro-symmetric and centro-skewsymmetric Toeplitz matrices and Bezoutians. Linear Algebra Appl., 343/344:195–209, 2002. Special issue on structured and inﬁnite systems of linear equations. [45] G. Heinig and K. Rost. Centrosymmetric and centro-skewsymmetric Toeplitz-plusHankel matrices and Bezoutians. Linear Algebra Appl., 366:257–281, 2003. Special issue on structured matrices: analysis, algorithms and applications (Cortona, 2000). [46] G. Heinig and K. Rost. Fast algorithms for centro-symmetric and centro-skewsymmetric Toeplitz-plus-Hankel matrices. Numer. Algorithms, 33(1-4):305–317, 2003. International Conference on Numerical Algorithms, Vol. I (Marrakesh, 2001). [47] G. Heinig and K. Rost. Split algorithms for Hermitian Toeplitz matrices with arbitrary rank proﬁle. Linear Algebra Appl., 392:235–253, 2004. [48] G. Heinig and K. Rost. Split algorithms for centrosymmetric Toeplitz-plus-Hankel matrices with arbitrary rank proﬁle. In The extended ﬁeld of operator theory, volume 171 of Oper. Theory Adv. Appl., pages 129–146. Birkh¨ auser, Basel, 2007. [49] C. Hermite. Extrait d’une lettre de Mr. Ch. Hermite de Paris `a Mr. Borchard de Berlin, sur le nombre des racines d’une ´equation alg´ebrique comprises entre des limites donn´ees. J. Reine Angew. Math., 52:39–51, 1856. [50] A.S. Householder. Bezoutiants, elimination and localization. SIAM Rev., 12:73–78, 1970. [51] C. Jacobi. De eliminatione variablis e duabus aequatione algebraicis. J. Reine Angew. Math., 15:101–124, 1836. [52] M.G. Krein and M.A. Naimark. The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations. Linear and Multilinear Algebra, 10(4):265–308, 1981. Translated from the Russian by O. Boshko and J.L. Howland. [53] B. Krishna and H. Krishna. Computationally eﬃcient reduced polynomial based algorithms for Hermitian Toeplitz matrices. SIAM J. Appl. Math., 49(4):1275–1282, 1989. [54] P. Lancaster and M. Tismenetsky. The theory of matrices. Computer Science and Applied Mathematics. Academic Press Inc., Orlando, FL, second edition, 1985. [55] F.I. Lander. The Bezoutian and the inversion of Hankel and Toeplitz matrices (in Russian). Mat. Issled., 9(2 (32)):69–87, 249–250, 1974. [56] A. Lascoux. Symmetric functions and combinatorial operators on polynomials, volume 99 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2003. [57] L. Lerer and L. Rodman. Bezoutians of rational matrix functions. J. Funct. Anal., 141(1):1–36, 1996. [58] L. Lerer and M. Tismenetsky. The Bezoutian and the eigenvalue-separation problem for matrix polynomials. Integral Equations Operator Theory, 5(3):386–445, 1982. [59] B. Mourrain and V.Y. Pan. Multivariate polynomials, duality, and structured matrices. J. Complexity, 16(1):110–180, 2000. Real computation and complexity (Schloss Dagstuhl, 1998).

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[60] A. Olshevsky and V. Olshevsky. Kharitonov’s theorem and Bezoutians. Linear Algebra Appl., 399:285–297, 2005. [61] V. Olshevsky and L. Sakhnovich. An operator identities approach to bezoutians. A general scheme and examples. In Proc.of the MTNS’ 04 Conference. 2004. [62] V.Y. Pan. Structured matrices and polynomials. Birkh¨ auser Boston Inc., Boston, MA, 2001. Uniﬁed superfast algorithms. [63] M.M. Postnikov. Ustoichivye mnogochleny (Stable polynomials). “Nauka”, Moscow, 1981. [64] K. Rost. Toeplitz-plus-Hankel Bezoutians and inverses of Toeplitz and Toeplitz-plusHankel matrices. Oper. Matrices, 2(3):385–406, 2008. [65] I. Sylvester. On a theory of syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s functions, and that of the greatest algebraical common measure. Phylos. Trans. Roy. Soc. London, 143:407–548, 1853. [66] H.K. Wimmer. On the history of the Bezoutian and the resultant matrix. Linear Algebra Appl., 128:27–34, 1990. Karla Rost Dept. of Mathematics Chemnitz University of Technology Reichenhainer Straße 39 D-09126 Chemnitz, Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 199, 119–123 c 2010 Birkh¨ auser Verlag Basel/Switzerland

On Matrices that are not Similar to a Toeplitz Matrix and a Family of Polynomials Tewodros Amdeberhan and Georg Heinig Abstract. A conjecture from the second author’s paper [Linear Algebra Appl., 332–334 (2001) 519–531] concerning a family of polynomials is proved and strengthened. A consequence of this is that for any n > 4 there is an n × n matrix that is not similar to a Toeplitz matrix, which was proved before for odd n and n = 6, 8, 10. Mathematics Subject Classiﬁcation (2000). Primary 15A21; Secondary 15A18. Keywords. Toeplitz matrix; Jordan normal form; Inverse eigenvalue problem.

1. Introduction In the paper [4] D.S. Mackey, N. Mackey and S. Petrovic posed and studied the inverse Jordan structure problem for complex Toeplitz matrices. They showed, in particular, that every n × n complex nonderogatory matrix is similar to an upper Hessenberg Toeplitz matrix, with ones on the subdiagonal. Such a choice guarantees uniqueness of the unit upper Hessenberg Toeplitz matrix. This result was recently extended by Willmer [6], who showed that a block companion matrix is similar to a unique block unit Hessenberg matrix. The authors [4] also investigated the problem of what happens if the nonderogatority condition is dropped and asked the question, “Is every complex matrix similar to a Toeplitz matrix?” This poses the inverse Jordan structure problem for Toeplitz matrices – which Jordan forms are achievable by Toeplitz matrices. Then, [4] gave an aﬃrmative answer to this question for matrices of order n ≤ 4 and conjectured that this might be true for all n. It is worth noting that the inverse eigenvalue question for real symmetric n × n Toeplitz matrices was posed in 1983 by Delsarte and Genin [1] and resolved by them for n ≤ 4; the general case was settled only recently by Landau [3]. Landau’s non-constructive proof uses topological degree theory to show that any list of n real numbers can be realized as the spectrum of an n × n real symmetric Toeplitz matrix.

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In [2] the second author of the present note showed that there are matrices that are not similar to a Toeplitz matrix. Examples for such matrices are m $

(S2 ⊕ c)

and

j=1

m−2 $

(S2 ⊕ S3 )

j=1

for all m > 1 and c = 0. Here Sk denotes the k × k i.e., ⎡

0 0 0 0 , S3 = ⎣ 1 0 S2 = 1 0 0 1

matrix of the forward shift, ⎤ 0 0 ⎦ , 0

and ⊕ stands for the direct sum. Note that the order of the ﬁrst set of these matrices is 2m + 1 and the second matrix is nilpotent. That means that for any odd integer n > 4 there is an n × n matrix that is not similar to a Toeplitz matrix. For even n the problem is more complicated. Candidates for matrices that are not similar to a Toeplitz matrix are m−1 $

(S2 ⊕ 0 ⊕ c)

j=1

and

m−2 $

(S2 ⊕ S3 ⊕ 0) ,

(1.1)

j=1

where c = 0 and m > 2. It was proved in [2] that these matrices are really not similar to a Toeplitz for m = 3, 4, 5, that means for matrices of order 6, 8 and 10. For the general case the problem was reduced to the property of a class of polynomials deﬁned as follows: 1 pj (t) = − pk (t)pj−k (t) 2 j−1

p0 (t) = p1 (t) = 1 ,

p2 (t) = t ,

(j > 2) .

(1.2)

k=1

It was shown that the matrices (1.1) are not similar to a Toeplitz matrix if the following is true. Condition 1.1 ([2], p. 528). For m > 3, the system of m − 2 equations pm+2 (t) = pm+3 (t) = · · · = p2m−1 (t) = 0 has only the trivial solution t = 0. In the present note we show that this condition is always satisﬁed. Even more, the following is shown, which is the main result of the paper. Theorem 1.2. For m > 1, pm+1 (t) = pm (t) = 0 has only the trivial solution t = 0. A consequence of this theorem is the following. Corollary 1.3. For any m > 4 there is an m × m matrix that is not similar to a Toeplitz matrix.

On Matrices that are not Similar to a Toeplitz Matrix

121

2. On a family of polynomials First we compute the generating function of the family of polynomials {pj (t)} deﬁned by (1.2), which is ∞ p(z, t) = pj (t)z j . j=0

Lemma 2.1. The generating function p(z, t) is given by 1/2 . p(z, t) = 1 + 2z + z 2 (2t + 1)

(2.1)

Proof. According to the deﬁnition of pj (t) we have pi (t)pk (t) = 0 i+k=j

for j > 2. That means that the coeﬃcients of z j in the expansion of (p(z, t))2 in powers of z vanish if j > 2. Hence p(z, t)2 is a quadratic polynomial in z, i.e., p(z, t) = A(t) + B(t)z + C(t)z 2 . Taking the deﬁnition of pj (t) for j = 0, 1, 2 into account we obtain A(t) = 1,

B(t) = 2,

C(t) = 2t + 1 ,

which completes the proof.

Expanding p(z, t) in powers of z we obtain the following explicit representation of pj (t)1 :

j/2 1/2 j−k (2t + 1)k , 2j−2k (2.2) pj (t) = j−k k k=0

where j/2 is the integer part of j/2. The key for proving Theorem 1.2 is the following lemma. Lemma 2.2. The polynomials pj (t) (j = 0, 1, . . . ) satisfy the 3-term recursion (j + 2)pj+2 (t) + (2j + 1)pj+1 (t) + (j − 1)(2t + 1)pj (t) = 0.

(2.3)

Proof. Let h(z, t) denote the generating function of the polynomial family {pj (t)} deﬁned by (2.3) with initial conditions p0 (t) = p1 (t) = 1. We show that h(z, t) = p(z, t). Let h denote the partial derivative of h(z, t) by z and h = h(z, t). We have ∞ ∞ (j + 2)pj+2 z j+1 = h − 1 , (2j + 1)pj+1 z j+1 = 2zh − h + 1 , j=0

j=0 ∞

(j − 1)pj z j+1 = z 2 h − zh .

j=0 1A

typo in [2] p. 528 is corrected here. The expression is never used to aﬀect the results of [2].

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Summing up we obtain the ordinary diﬀerential equation (1 + 2z + (2t + 1)z 2 )h − (1 + (2t + 1)z)h = 0 . As it is easily checked, the generating function p(z, t) also satisﬁes this equation. Since p(0, t) = h(0, t), we conclude that p(z, t) = h(z, t). An alternative way to prove the lemma is to employ the explicit expression (2.2) for pj (t). This appears in the Appendix section. Proof of Theorem 1.2. The theorem can be proved now by induction in a standard fashion. The base case, m = 2, is evident since p2 = t = p3 = 0 iﬀ t = 0. Assume the theorem is valid for m > 1, then we claim the same is true for m + 1. Suppose not! i.e., pm+2 (τ ) = pm+1 (τ ) = 0 for some τ = 0. Then Lemma 2.2 implies that τ = − 12 . Once again, make application of the recurrence (2.3) but this time reindex m by m − 1 to get (m + 1)pm+1 (τ ) + (2m − 1)pm (τ ) + (m − 2)(2τ + 1)pm−2 (τ ) = 0. pm (− 12 )

So, = 0. Hence both pm+1 and pm vanish at the induction step proves the theorem.

− 21 .

(2.4)

This contradiction to

Let us ﬁnally mention two consequences of our result. The following is imb mediate from Theorem 1.2 where variables are switched w = 2a z and the value 4ac t = b2 − 1 is selected. The case b = 0 is treated separately. It is important that t = 0. 1

Corollary 2.3. Let f (w) = (a + bw + cw2 ) 2 , where a = 0 and b2 − 4ac = 0, and ∞ f (w) = k=0 fk z k be its Maclaurin expansion. Then for all j, fj and fj+1 cannot both vanish. The following is an equivalent formulation of Condition 1.1. Corollary 2.4. For n > 4 there is no polynomial P (t) of degree n such that P (t)2 = q(t) + t2n−1 r(t) for quadratic polynomials q(t) and r(t), except for the trivial cases P (t) = a + bt and P (t) = atn−1 + btn . Proof. Compare proof of Lemma 6.1 in [2] where the polynomials pj (t) take the place of uk . Then, convert uk via uk /uk1 .

3. Appendix We show a scheme on how to arrive at the recursion (j + 2)pj+2 (t) + (2j + 1)pj+1 (t) + (j − 1)(2t + 1)pj (t) = 0 for the explicit expression

j/2

pj (t) =

k=0

2j−2k

1/2 j−k

j−k (2t + 1)k k

(3.1)

On Matrices that are not Similar to a Toeplitz Matrix

123

of the sequence {pj (t)}j . The idea utilizes the so-called Wilf-Zeilberger (WZ) method of proof [5]. 1/2 j−k (j−1)(2j−2k−1)k k Let F (j, k) := 2j j−k k (2t + 1) , and G(j, k) := −2 (j+1−2k)(j+2−2k) F (j, k). Then one can check, preferably using a symbolic software, that (j + 2)F (j + 2, k)+ (2j + 1)F (j + 1, k)+ (j − 1)(2t+ 1)F (j, k) = G(j, k + 1)− G(j, k). Telescoping: Sum over all −∞ < k < ∞ and observe that ∞

j/2

F (j, k) =

k=−∞

F (j, k) = pj (t)

k=0

while

∞

G(j, k + 1) =

k=−∞

∞

G(j, k),

k=−∞

since G(j, k) has compact support. Then assertion (3.1) follows. Acknowledgment The ﬁrst author gratefully acknowledges the wonderful support rendered, at onset of this project, by the DIMACS center at Rutgers University. He also takes this opportunity to commemorate the second author as a great Mensch.

References [1] Philippe Delsarte, Yves V. Genin. Spectral properties of ﬁnite Toeplitz matrices. in Mathematical Theory of networks and Systems: Proc. MTNS-83 Int. Symp., Beer Sheva, Israel, June 1983, P.A. Fhrmann, ed., Lecture Notes in Control and Information Sciences, New York, Springer-Verlag, 58:194–213, 1984. [2] Georg Heinig. Not every matrix is similar to a Toeplitz matrix. Linear Algebra Appl., 332–334:519–531, 2001. [3] Henry J Landau. The inverse eigenvalue problem for real symmetric Toeplitz matrices. J. Amer. Math. Soc., 7:749–767, 1994. [4] D. Steven Mackey, Niloufer Mackey, and Srdjan Petrovic. Is every matrix similar to a Toeplitz matrix? Linear Algebra Appl. 297:87–105, 1999. [5] Marko Petkovsek, Herbert Wilf, Doron Zeilberger. A = B Toeplitz matrices. A.K. Peters Ltd., USA, 1996. [6] Harald K. Wimmer. Similarity of block companion and block Toeplitz matrices. Linear Algebra Appl. 343–344:381–387, 2002. Tewodros Amdeberhan Mathematics Tulane University New Orleans, LA 70118, USA e-mail: [email protected]

Part II Research Contributions

Operator Theory: Advances and Applications, Vol. 199, 127–154 c 2010 Birkh¨ auser Verlag Basel/Switzerland

A Traub-like Algorithm for Hessenbergquasiseparable-Vandermonde Matrices of Arbitrary Order T. Bella, Y. Eidelman, I. Gohberg, V. Olshevsky, E. Tyrtyshnikov and P. Zhlobich Abstract. Although Gaussian elimination uses O(n3 ) operations to invert an arbitrary matrix, matrices with a special Vandermonde structure can be inverted in only O(n2 ) operations by the fast Traub algorithm. The original version of Traub algorithm was numerically unstable although only a minor modiﬁcation of it yields a high accuracy in practice. The Traub algorithm has been extended from Vandermonde matrices involving monomials to polynomial-Vandermonde matrices involving real orthogonal polynomials, and the Szeg¨ o polynomials. In this paper we consider a new more general class of polynomials that we suggest to call Hessenberg order m quasiseparable polynomials, or (H, m)quasiseparable polynomials. The new class is wide enough to include all of the above important special cases, e.g., monomials, real orthogonal polynomials and the Szeg¨ o polynomials, as well as new subclasses. We derive a fast O(n2 ) Traub-like algorithm to invert the associated (H, m)-quasiseparableVandermonde matrices. The class of quasiseparable matrices is garnering a lot of attention recently; it has been found to be useful in designing a number of fast algorithms. The derivation of our new Traub-like algorithm is also based on exploiting quasiseparable structure of the corresponding Hessenberg matrices. Preliminary numerical experiments are presented comparing the algorithm to standard structure ignoring methods. This paper extends our recent results in [6] from the (H, 0)- and (H, 1)quasiseparable cases to the more general (H, m)-quasiseparable case. Mathematics Subject Classiﬁcation (2000). 15A09, 15-04, 15B05. Keywords. Orthogonal polynomials, Szego polynomials, quasiseparable matrices, Vandermonde matrices, Hessenberg matrices, inversion, fast algorithm.

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1. Introduction. Polynomial-Vandermonde matrices and quasiseparable matrices 1.1. Inversion of polynomial-Vandermonde matrices In this paper we consider the problem of inverting the class of polynomial-Vandern monde matrices. a set For of n distinct nodes {xk }k=1 , the classical Vandermonde j−1 matrix V (x) = xi is known to be invertible (provided the nodes are distinct). One can generalize this structure by evaluating a diﬀerent basis (other than the monomials) at the nodes in the following way. That is, for a set of n polynomials R = {r0 (x), r1 (x), . . . , rn−1 (x)} satisfying deg rk (x) = k, the matrix of the form ⎤ ⎡ r0 (x1 ) r1 (x1 ) · · · rn−1 (x1 ) ⎢ r0 (x2 ) r1 (x2 ) · · · rn−1 (x2 ) ⎥ ⎥ ⎢ VR (x) = ⎢ (1.1) ⎥ .. .. .. ⎦ ⎣ . . . r0 (xn ) r1 (xn )

· · · rn−1 (xn )

is called a polynomial-Vandermonde matrix. It is clear that such a matrix is invertible if and only if the chosen nodes are distinct. Indeed, let T be an invertible, upper triangular matrix, and consider the product V (x) · T . The eﬀect of postmultiplication by T is that the entries of the product are polynomials determined by the columns of T , evaluated at the given nodes. Hence this is an alternate definition of a polynomial-Vandermonde matrix as the product of invertible matrices (provided the nodes are distinct). In the simplest case where R = {1, x, x2 , . . . , xn−1 } (i.e., when T = I), the matrix VR (x) reduces to a classical Vandermonde matrix and the inversion algorithm is due to Traub [28]. It was observed in [16] that a minor modiﬁcation of the original Traub algorithm results in very good accuracy. The structure-ignoring approach of Gaussian elimination for inversion of VR (x) requires O(n3 ) operations, and for a general matrix VR (x) (i.e., no special recurrence relations satisﬁed by the polynomial system R involved), the algorithm derived in this paper also requires O(n3 ) operations. However, in several special cases, the structure has been exploited, resulting in fast algorithms that can compute the n2 entries of the inverse in only O(n2 ) operations. It also allows the construction of fast system solvers; one of the pioneering works in this area belongs to Bj¨orck and Pereyra [2]. Table 1 lists the previous work in deriving fast inversion algorithms and fast system solvers for various special cases of the polynomial system R. 1.2. Capturing recurrence relations via confederate matrices To generalize the inversion algorithms of Table 3 we will use the concept of a confederate matrix introduced in [21]. Let polynomials R = {r0 (x), r1 (x) , . . . , rn (x)} be speciﬁed by the general n-term recurrence relations1 rk (x) = (αk x−ak−1,k )·rk−1 (x)−ak−2,k ·rk−2 (x)−· · ·−a0,k ·r0 (x), 1 It

αk = 0 (1.2)

is easy to see that any polynomial system {rk (x)} satisfying deg rk (x) = k obeys (1.2).

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Table 1. Fast O(n2 ) algorithms for polynomial-Vandermonde matrices. Matrix VR (x)

Polynomial System R O(n2 ) inversion

O(n2 ) system solver

Classical Vdm

monomials

Traub [28]

Bj¨ orck–Pereyra [2]

Chebyshev–Vdm

Chebyshev plns

Gohberg–Ols [14]

Reichel–Opfer [26]

Three-term Vdm Real orthogonal plns

Calvetti–Reichel [7] Higham [19]

Szeg¨ o–Vdm

Olshevsky [23]

Szeg¨ o plns

Vdm = Vandermonde;

Ols = Olshevsky;

BEGKO [3]

plns = polynomials

for k > 0, and r0 is a constant. Deﬁne for the polynomial P (x) = P0 · r0 (x) + P1 · r1 (x) + · · · + Pn−1 · rn−1 (x) + Pn · rn (x) its confederate matrix (with respect to the polynomial system R) by ⎡ a01 a02 a03 a a0,n ⎤ · · · α0,k ··· ··· α1 α2 α3 αn k a ⎢ α1 aα12 aα13 · · · α1,k · · · · · · aα1,n ⎥ n k ⎥ ⎡ ⎢ 1 2 3 ⎤ ⎥ ⎢ .. P0 a2,n ⎥ ⎢ 0 1 a23 · · · . ··· ··· ⎥ ⎢ P1 ⎥ ⎢ α2 α3 αn ⎥ ⎢ ⎢ .. ⎥ ⎥ ⎢ 0 0 1 . . . ak−2,k . . . ⎢ . ⎥ ⎥ ⎢ P2 ⎥ ⎢ α3 αk ⎥ ⎢ ⎢ CR (P ) = ⎢ . .. ⎥ − ⎢ ... ⎥ .. . . . . ak−1,k . . . . ⎥ 0 ··· 0 . . . . α . ⎥ ⎢ . ⎥ ⎢ .. k ⎥ ⎢ . ⎥ ⎢ . .. . . . . . ⎥ ⎢ .. .. .. .. ⎥ ⎣ . ⎦ ⎢ . . . α1 . . . k ⎥ ⎢ Pn−1 ⎢ .. .. ⎥ .. . . . . .. .. .. ⎣ . . . . . . . ⎦ . an−1,n 1 0 0 · · · · · · · · · 0 αn−1 αn CR (rn )

(1.3)

1 αn P n

(1.4)

In the special case where P (x) = rn (x), we have P0 = P1 = · · · = Pn−1 = 0, and hence the last term on the right-hand side of (1.4) vanishes. Notice that the coeﬃcients of the recurrence relations for the k th polynomial r (x) from (1.2) are contained in the k th column of C (r ), as the highlighted k

R

n

column shows. We refer to [21] for many useful properties of the confederate matrix and only recall here that rk (x) = α0 · α1 · . . . · αk · det(xI − [CR (P )]k×k ), and P (x) = α0 · α1 · · · αn · Pn · det(xI − CR (P )), where [CR (P )]k×k denotes the k × k leading submatrix of CR (P ) in the special case where P (x) = rn (x).

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Next in Table 2 we list confederate matrices for the polynomial systems2 of Table 3. Table 2. Systems of polynomials and corresponding recurrence relations. Polynomial System Recurrence relations

Corresponding confederate matrix CR (rn ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

monomials rk (x) = x · rk−1 (x)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Chebyshev polynomials rk (x) = 2x · rk−1 (x) − rk−2 (x)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

real orthogonal polynomials rk (x) = (αk x − δk )rk−1 (x) −γk · rk−2 (x)

0

0

1

..

.

..

.

0 .. . 0

..

. ··· lower 1 2

··· ··· 0 .. . . 1 . . 0 2 .. .. .. .. . . . . 0 . .. .. .. .. . 1 . . 2 1 0 ··· 0 0 2 tridiagonal matrix

0

δ1 α1

γ2 α2

1 α1

δ2 α2

0

1 α2

.. .

..

0

··· ··· 0 .. . 0 . .. .. . .. . . .. .. . .. . 0 1 0 shift matrix

0

···

..

.

..

..

.

.

..

.

···

0

.

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0 .. .

γn−1 αn−1

0

δn−1 αn−1 1 αn−1

γn αn δn αn

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

tridiagonal matrix ⎡ Szeg¨ o polynomials3

φk (x) 1 −ρ∗k = μ1 # k φk (x) % −ρk 1 & φk−1 (x) × xφ# k−1 (x)

−ρ1 ρ∗0 · · · −ρn−1 μn−2 . . . μ1 ρ∗0 −ρn μn−1 . . . μ1 ρ∗0 ⎢ .. ⎢ . −ρn−1 μn−2 . . . μ2 ρ∗1 −ρn μn−1 . . . μ2 ρ∗1 ⎢ μ1 ⎢ ⎢ . . .. .. .. ⎢ . 0 ⎢ ⎢ .. ⎢ ⎣ −ρn μn−1 ρ∗n−2 . −ρn−1 ρ∗n−2 0 ··· μn−1 −ρn ρ∗n−1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

unitary Hessenberg matrix matrix 2 For the monomials, Chebyshev polynomials and real orthogonal polynomials the structure of the confederate matrices can be immediately deduced from their recurrence relations. For Szeg¨o polynomials it is also well known, see, e.g., [23] and the references therein.

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It turns out that all matrices of Table 2 are special cases of the more general class of matrices deﬁned next. It is this larger class of matrices, and the class of polynomials related to them via (1.2) that we consider in this paper. 1.3. Main tool: quasiseparable matrices and polynomials Deﬁnition 1.1. (Quasiseparable matrices and polynomials) • A matrix A is called (H, m)-quasiseparable (i.e., Hessenberg lower part and order m upper part) if (i) it is strongly upper Hessenberg (i.e., nonzero ﬁrst subdiagonal, ai+1,i = 0), and (ii) max(rank A12 ) = m, where the maximum is taken over all symmetric partitions of the form

∗ A12 A= ∗ ∗ • Let A = [aij ] be a (H, m)-quasiseparable matrix. For αi = 1/ai+1,i , then the system of polynomials related to A via rk (x) = α1 · · · αk det (xI − A)(k×k) . is called a system of (H, m)-quasiseparable polynomials. Remark 1.2. The class of (H, m)-quasiseparable polynomials is wide enough to include monomials, Chebyshev polynomials, real orthogonal and Szeg¨ o polynomials (i.e., all polynomials of Tables 3 and 2) as special cases. This can be seen by inspecting, for each confederate matrix, its typical submatrix A12 from the partition described in Deﬁnition 1.1. • The lower shift matrix is (H, 0)-quasiseparable Indeed, if A is such a matrix, then any submatrix A12 is simply a zero matrix. • Tridiagonal matrices are also (H, 1)-quasiseparable. Indeed, if A is tridiagonal, then the submatrix A12 has the form (γj /αj )ek eT1 , which can easily be observed to have rank one. • Unitary Hessenberg matrices are (H, 1)-quasiseparable. Indeed, if A corresponds to the Szeg¨o polynomials, then the corresponding 3×(n−1) submatrix A12 has the form ⎡ ⎤ −ρk μk−1 · · · μ3 μ2 μ1 ρ∗0 −ρk−1 μk−2 · · · μ3 μ2 μ1 ρ∗0 · · · −ρn μn−1 · · · μ3 μ2 μ1 ρ∗0 ⎣ −ρk μk−1 · · · μ3 μ2 ρ∗1 −ρk−1 μk−2 · · · μ3 μ2 ρ∗1 · · · −ρn μn−1 · · · μ3 μ2 ρ∗1 ⎦, −ρk μk−1 · · · μ3 ρ∗2 −ρk−1 μk−2 · · · μ3 ρ∗2 · · · −ρn μn−1 · · · μ3 ρ∗2 which is also rank 1 since the rows are scalar multiples of each other. The same is true for all other symmetric partitions of A. 3 It

is known that, under the additional restriction of ρk = 0 for each k, the corresponding Szeg¨ o polynomials satisfy the three-term recurrence relations φ# 0 (x) =

φ# k (x)

=

1

'μ0

,

1 μk

φ# 1 (x) = ·x+

# 1 ∗ (x · φ# 0 (x) + ρ1 ρ0 · φ0 (x)) μ1 ( # ρk μk−1 1 φk−1 (x) − ρ ·x· ρk−1 μk k−1 μk ρk

φ# k−2 (x).

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Hence all of the polynomials corresponding to the confederate matrices listed above are (H, 1)-quasiseparable polynomials. 1.4. Main problem: Inversion of (H, m)-quasiseparable-Vandermonde matrices As shown in the previous remark, (H, 0)- and (H, 1)-quasiseparable polynomials generalize the previous cases of monomials, real orthogonal polynomials, and Szeg¨o polynomials. In the paper [6], an algorithm for inversion of (H, 0)- and (H, 1)-quasiseparable-Vandermonde matrices is derived, and hence that algorithm is applicable to these special cases. However there are important cases not covered by either (H, 0)- or (H, 1)-quasiseparable polynomials. An example of such a system of polynomials is given next. Example 1.3 (l-recurrent polynomials). From (1.4) it follows that if polynomials satisfy l-term recurrence relations rk (x) = (αk x−ak−1,k )·rk−1 (x)−ak−2,k ·rk−2 (x)−· · ·−ak−(l−1),k ·rk−(l−1) (x) (1.5) then their confederate matrices ⎡ a0,1 ··· α1 ⎢ 1 a1,2 ⎢ ⎢ α1 α2 ⎢ ⎢ 0 1 ⎢ α2 A=⎢ . .. ⎢ . . ⎢ . ⎢ ⎢ .. ⎣ . 0 ···

a0,l−1 αl−1

0

···

a1,l αl

..

..

.

..

0

⎤

0 .. . 0 an−(l−1),n αn

.

1 αn−2

. ···

··· .. . .. .

.. .

1 αn−1

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(1.6)

an−1,n αn

are (1, l−2)-banded, i.e., they have only one nonzero subdiagonal and l−2 nonzero superdiagonals. Considering a typical element A12 of the partition of Deﬁnition 1.1, in this case for a 5 × 5, (1, 2)-banded example, we have ⎤ ⎡ a0,1 a0,2 a0,3 0 0 α1 α2 α3 a1,2 a1,3 a1,4

⎢ α11 0 ⎥ α2 α3 α4 ⎢ ∗ A12 a a2,4 a2,5 ⎥ 2,3 1 ⎥ 0 A= =⎢ α2 α3 α4 α5 ⎥ ⎢ ∗ ∗ a3,4 a3,5 1 ⎦ ⎣ 0 0 0

0

α3

0

α4 1 α4

α5 a4,5 α5

One can see that any A12 of the partition of Deﬁnition 1.1 has rank at most 2, implying that A is a (H, 2)-quasiseparable matrix by deﬁnition. More generally, a system of l-recurrent polynomials are (H, l−2)-quasiseparable (and so polynomials satisfying three-term recurrence relations are (H, 1)-quasiseparable). A Bj¨orck–Pereyra-like algorithm for solving linear systems with Hessenbergquasiseparable-Vandermonde coeﬃcient matrices was proposed in [4], and this algorithm is applicable for any order of quasiseparability (i.e., for any m 1 of (H, m)-quasiseparable matrix). A Traub-like inversion algorithm is derived in [6], but it is valid only for (H, 0)- and (H, 1)-quasiseparable-Vandermonde matrices.

A Traub-like Algorithm

133

In this paper we extend the Traub-like algorithm to a corresponding algorithm for an arbitrary order of quasiseparability. Previous work in this area, including that using quasiseparable matrices, is given next in Table 3. Table 3. Fast O(n2 ) algorithms for polynomial-Vandermonde matrices. Matrix VR (x)

Polynomial System R O(n2 ) inversion

O(n2 ) system solver

Classical Vdm

monomials

Traub [28]

Bj¨ orck–Pereyra [2]

Chebyshev–Vdm

Chebyshev plns

Gohberg–Ols [14]

Reichel–Opfer [26]

Three-term Vdm

Real orthogonal plns

Calvetti–Reichel [7] Higham [19]

Szeg¨ o–Vdm

Szeg¨ o plns

Ols [23]

BEGKO [3]

(H, 1)-qsep-Vdm

(H, 1)-qsep

BEGOT [6]

BEGKO [4]

(H, m)-qsep-Vdm (H, m)-qsep

this paper

Vdm = Vandermonde; Ols = Olshevsky; plns = polynomials; qsep = quasiseparable

This new inversion algorithm is applicable to the special cases of polynomialVandermonde matrices for monomials, real orthogonal polynomials, and Szeg¨ o polynomials, which are themselves special cases of (H, m)-quasiseparable polynomials. In addition, it is also applicable to new classes of polynomials for which no Traub-like algorithm is currently available. One such class of polynomials are those satisfying the motivating recurrence relations (1.5). As was the case for the Traub-like algorithm for (H, 0)- and (H, 1)-quasiseparable-Vandermonde matrices of [6], the proposed Traub-like algorithm for (H, m)quasiseparable-Vandermonde matrices is fast, requiring only O(n2 ) operations by exploiting the sparse recurrence relations (1.5).

2. Inversion formula In this section we recall the formula that will be used to invert a polynomialVandermonde matrix as in (1.1). Such a matrix is completely determined by n polynomials R = {r0 (x), . . . , rn−1 (x)} and n nodes x = (x1 , . . . , xn ). The desired inverse VR (x)−1 is given by the formula VR (x)−1 = I˜ · VR)T (x) · diag(c1 , . . . , cn ),

(2.1)

(see [22], [23]) where ci =

n k=1

k =i

(xk − xi )−1 ,

(2.2)

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I˜ is the antidiagonal matrix

⎡

⎤ 0 ··· 0 1 ⎢ .. ⎥ ⎢ . ... 1 0 ⎥ ˜ ⎢ ⎥ I=⎢ . ⎥, ⎣ 0 . . . . . . .. ⎦ 1 0 ··· 0

(2.3)

) is the system of associated (generalized Horner) polynomials, deﬁned as and R follows: if we deﬁne the master polynomial P (x) by P (x) = (x − x1 ) · · · (x − xn ), then for the polynomial system R = {r0 (x), . . . , rn−1 (x), P (x)}, the associated ) = {) polynomials R r0 (x), . . . , r)n−1 (x), P (x)} are those satisfying the relations n−1 P (x) − P (y) rk (x) · r)n−k−1 (y), = x−y

(2.4)

k=0

see [20]. A discussion showing the existence of polynomials satisfying these relations (2.4) for any polynomial system R is given in [3]. This deﬁnition can be seen as a generalization of the Horner polynomials associated with the monomials, cf. with the discussion in Section 2.1 below. This discussion gives a relation between the inverse VR (x)−1 and the polyno) is the system of polynomials associated mial-Vandermonde matrix VR) (x), where R with R. To use this in order to invert VR (x), one needs to evaluate the polynomials ) at the nodes x to form V T (x). Such evaluation can be done by using confederate R ) R matrices (deﬁned in Section 1.2) associated with system of polynomials, which will be discussed in the next section, but at this point the formula (2.1) allows us to present a sketch of the Traub-like inversion algorithm next. The detailed algorithm will be provided in Section 6 below after deriving next several formulas that will be required to implement its steps 2 and 3. Algorithm 2.1 (A sketch of the Traub-like inversion algorithm). 1. Compute the entries of diag(c1 , . . . , cn ) via (2.2). 2. Compute the coeﬃcients {P0 , P1 , . . . , Pn } of the master polynomial P (x) as in (1.3). ) with confederate matrix speciﬁed via (2.8) 3. Evaluate the n polynomials of R at the n nodes xk to form VR) (x). 4. Compute the inverse VR (x)−1 via (2.1). 2.1. The key property of all Traub-like algorithms: pertransposition In this section we use the classical Traub algorithm to explain the key property used in deriving a Traub-like algorithms in terms of quasiseparable ranks of confederate matrices of systems of polynomials. According to (1.3), let P (x) = P0 + P1 · x + · · · + Pn−1 · xn−1 + xn be a polynomial in the monomial base. The monomials R = {1, x, x2 , . . . , xn−1 } satisfy the obvious recurrence relations xk = x · xk−1 and hence the confederate

A Traub-like Algorithm matrix (1.4) of P (x) with respect to ⎡ 0 ⎢ 1 ⎢ ⎢ CR (P ) = ⎢ ⎢ 0 ⎢ . ⎣ ..

135

R becomes 0 0 1 ..

. 0 ···

··· 0 ··· 0 . .. . .. .. . 0 0

1

−P0 −P1 .. . .. . −Pn−1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(2.5)

which is the well-known companion matrix. By Deﬁnition 1.1 its leading submatrices [CR (P )]k×k are (H, 0)-quasiseparable for k = 1 . . . n − 1. Hence monomials are (H, 0)-quasiseparable polynomials. From the well-known recurrence relations for the Horner polynomials (which invert the classical Vandermonde matrix, see [28]) r)0 (x) = 1,

r)k (x) = x · r)k−1 (x) + Pn−k ,

we obtain the confederate matrix ⎡ −Pn−1 ⎢ 1 ⎢ ⎢ 0 CR) (P ) = ⎢ ⎢ ⎢ .. ⎣ . 0

−Pn−2 0 1 .. . ···

· · · −P1 ··· 0 .. .. . . .. . 0 0

1

(2.6)

⎤ −P0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥. .. ⎥ . ⎦ 0

(2.7)

for the Horner polynomials. This relation between the confederate matrices of R ) can be seen as and R ˜ CR) (P ) = I˜ · CR (P )T · I,

(2.8)

or more visually,

a

b Pertransposition

-

CR (P ) = b

@ I @ @ R @

= CR) (P ), a

and holds in the general case (see [22], [23]). The passage from CR (P ) to CR) (P ) in (2.8) is called a pertransposition, or reﬂection across the antidiagonal. We will show later that recurrence relations for a given system of polynomials together ) at the nodes x, a required with (2.8) allow fast evaluation of the polynomials R step for a fast Traub-like algorithm.

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The leading submatrices of the matrix (2.7) are easily seen to be Hessenberg and quasiseparable4 but due to the perturbation {P0 , P1 , . . . , Pn−1 } their quasiseparable ranks are 1. Hence the Horner polynomials are (H, 1)-quasiseparable; that is, the quasiseparable rank increases by one due to the inclusion of the perturbation terms in each principal submatrix. Analogously, for an arbitrary system of (H, m)-quasiseparable polynomials the order of quasiseparability increases by at most one, which we state as follows. Remark 2.2. For R a system of (H, m)-quasiseparable polynomials, when passing ) of polynomials associated with R, the order of quasiseparability to the system R ) is either (H, m)-quasiseparable or may increase by one. That is, the system R (H, m + 1)-quasiseparable. This property is used by all of the previous Traub-like algorithms given in Table 3. The quasiseparable rank of a confederate matrix after pertransposition is increased only by at most one. This allows derivation of cheap recurrence relations ) and use them in computing V ) . for the system of polynomials R R In summary, the classical Traub algorithm was based on deriving formulas for systems of polynomials as the confederate matrices changed from (H, 0)-quasiseparable to (H, 1)-quasiseparable, and subsequent Traub-like algorithms were based on changes from (H, 1)-quasiseparable to (H, 2)-quasiseparable. Such derivations were already very involved. In order to derive a Traub-like algorithm in the more general case considered in this paper, we need to (i) attain the formulas for the original (H, m)quasiseparable polynomials which, unlike previous cases, are not readily available5 , and (ii) have a derivation allowing us to pass from (H, m)-quasiseparable to (H, m + 1)-quasiseparable confederate matrices. Hence our plan for the next two sections is to solve the problems (i) and (ii) listed above, respectively.

3. Recurrence relations for (H, m)-quasiseparable polynomials Real orthogonal polynomials are typically deﬁned in terms of the recurrence relations they satisfy, and then these recurrence relations are used to give equivalent deﬁnitions in terms of Hessenberg matrices, as in Table 2. Currently we have only the latter deﬁnition for (H, m)-quasiseparable polynomials, i.e., in terms of the related (H, m)-quasiseparable matrix. Since the main tool is designing fast algorithms is the recurrence relations, as in (2.6), it is the goal of this section to derive a set of sparse recurrence relations satisﬁed by (H, m)-quasiseparable polynomials. 4 Pertransposition

changes the order in which the submatrices A12 of Deﬁnition 1.1 appear and transposes them, but does not change their ranks, hence the quasiseparability is preserved. 5 In the (H, 1)-quasiseparable case, these formulas were derived in [6].

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3.1. Generators of quasiseparable matrices We begin with an equivalent deﬁnition of quasiseparability in terms of generators. Such generators are the compressed representation of a quasiseparable matrix; that is, the O(n) entries of the generators deﬁne the O(n2 ) entries of the matrix. Operations with generators are the key in designing various types of fast algorithms. In this case, the sparse recurrence relations for (H, m)-quasiseparable polynomials will be given in terms of these generators. Deﬁnition 3.1 (Generator deﬁnition for (H, m)-quasiseparable matrices). A matrix A is called (H, m)-quasiseparable if (i) it is strongly upper Hessenberg (i.e., nonzero ﬁrst subdiagonal, ai+1,i = 0), and (ii) it can be represented in the form @ d @ 1@ g i b× @ @ ij hj . . p2 q1@ .@ @ @ A= @ @ @ @ . @ .. @ @ @ .@ .. @ @ @ @ @ @ 0 p q n n−1 @ @ dn@ @ @ where

b× ij

= bi+1 · · · bj−1 for j > i + 1 and

b× ij

(3.1)

= 1 for j = i + 1. The elements

{pk , qk , dk , gk , bk , hk }, called the generators of the matrix A, are matrices of sizes pk qk dk gk bk hk sizes 1×1 1×1 1×1 1 × uk uk−1 × uk uk−1 × 1 range k ∈ [2, n] k ∈ [1, n − 1] k ∈ [1, n] k ∈ [1, n − 1] k ∈ [2, n − 1] k ∈ [2, n] subject to maxk uk = m. Remark 3.2. For a given (H, m)-quasiseparable matrix the set of generators of Deﬁnition 3.1 is not unique. There is a freedom in choosing generators without changing the matrix. Remark 3.3. It is useful to note that Deﬁnition 3.1 together with (1.2) imply that (H, m)-quasiseparable polynomials satisfy n-term recurrence relations ⎤ ⎡ k−2 1 ⎦. ⎣(x − dk )rk−1 (x) − (3.2) rk (x) = gj+1 b× j+1,k hk rj (x) pk+1 qk j=0

This formula is not sparse and hence expensive. In order to design a fast algorithm, sparse recurrence relations are required, and such are stated and proved in the next section.

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3.2. Sparse recurrence relations for (H, m)-quasiseparable polynomials The next theorem gives, for any (H, m)-quasiseparable matrix, a set of sparse recurrence relations satisﬁed by the corresponding (H, m)-quasiseparable polynomials. These recurrence relations are given in terms of the generators of the (H, m)-quasiseparable matrix. Theorem 3.4. Let A be a (H, m)-quasiseparable matrix speciﬁed by the generators {pk , qk , dk , gk , bk , hk }. Then the polynomial system R = {rk (x)}nk=0 corresponding to A (such that A = CR (rn )) satisﬁes ⎡ ⎤⎡ ⎤ ⎤ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Fk (x)

rk (x)

⎥ ⎥ ⎥ 1 ⎥ ⎥= ⎥ pk+1 qk ⎥ ⎦

⎢ ⎢ ⎢ pk qk ⎢ ⎢ ⎢ ⎢ ⎣ pk

bTk

−qk gkT

hTk

x − dk

⎥⎢ ⎥⎢ ⎥ ⎢ Fk−1 (x) ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣ rk−1 (x)

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.3)

The proof is given at the end of this section, however ﬁrst some special cases are given in detail. Example 3.5 ((H, 0)- and (H, 1)-quasiseparable case). For the case where m 1, the recurrence relations (3.3) reduce to those derived in [11], which were used in [6] to derive the Traub-like algorithm for (H, 0)- and (H, 1)-quasiseparableVandermonde matrices; that is, of the form

αk βk Fk−1 (x) Fk (x) = . (3.4) rk (x) rk−1 (x) γk δk x + θk These were referred to as [EGO05]-type recurrence relations in [6], and as the recurrence relations (3.3) are a generalization of these, we refer to (3.3) as [EGO05]type recurrence relations as well. A motivating example for considering the larger class of (H, m)-quasiseparable polynomials was their inclusion of the class of l-recurrent polynomials, a class not contained by previous cases. Example 3.6 (l-recurrent polynomials). By introducing the auxiliary polynomials (1) (l−2) fk (x), . . . , fk (x), the relation (1.5) can be rewritten as ⎤ ⎤ ⎡ ⎡ (1) ⎤ ⎡ (1) fk−1 (x) 0 1 0 ... 0 −ak−2,k fk (x) ⎥ ⎥ ⎢ . . ⎢ ⎥⎢ .. .. .. ⎥ ⎥ ⎢ . ⎢ . . . . . . . . ... ⎥⎢ . . . ⎢ ⎥ ⎥ ⎢ . ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥⎢ .. .. .. .. .. .. ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ . . 0 . . . ⎢ ⎥, ⎥=⎢ . ⎢ ⎥⎢ ⎥ ⎥ ⎢ ⎥ ⎢ .. .. . .. ⎢ ⎥ ⎥ ⎢ .. ⎢ .. ⎥⎢ . 1 ⎥ ⎥ ⎢ . ⎢ . . ⎥ ⎢ ⎥ ⎢ (l−2) ⎥ ⎣ ⎦ ⎣ f 0 . . . . . . . . . 0 −ak−(l−1),k ⎦ ⎣ f (l−2) ⎦ k k−1 1 0 . . . . . . 0 αk x − ak−1,k rk (x) rk−1 (x) which is the reduction of (3.3) in this special case.

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Proof of Theorem 3.4. The recurrence relations (3.3) deﬁne a system of polynomials which satisfy the n-term recurrence relations rk (x) = (αk x − ak−1,k ) · rk−1 (x) − ak−2,k · rk−2 (x) − · · · − a0,k · r0 (x)

(3.5)

for some coeﬃcients αk , ak−1,k , . . . , a0,k . The proof is presented by showing that these n-term recurrence relations in fact coincide exactly with (3.2), so these coefﬁcients coincide with those of the n-term recurrence relations of the polynomials R. Using relations for rk (x) and Fk−1 (x) from (3.3), we have rk (x) =

1 pk+1 qk

(x − dk )rk−1 (x) − gk−1 hk rk−2 (x) + pk−1 hTk bTk−1 Fk−2 (x) .

(3.6) Notice that again using (3.3) to eliminate Fk−2 (x) from the equation (3.6) will result in an expression for rk (x) in terms of rk−1 (x), rk−2 (x), rk−3 (x), Fk−3 (x), and r0 (x) without modifying the coeﬃcients of rk−1 (x), rk−2 (x), or r0 (x). Again applying (3.3) to eliminate Fk−3 (x) results in an expression in terms of rk−1 (x), rk−2 (x), rk−3 (x), rk−4 (x), Fk−4 (x), and r0 (x) without modifying the coeﬃcients of rk−1 (x), rk−2 (x), rk−3 (x), or r0 (x). Continuing in this way, the n-term recurrence relations of the form (3.5) are obtained without modifying the coeﬃcients of the previous ones. Suppose that for some 0 < j < k − 1 the expression for rk (x) is of the form 1 ' rk (x) = (x − dk )rk−1 (x) − gk−1 hk rk−2 (x) − · · · pk+1 qk (3.7) ( T × T · · · − gj+1 b× h r (x) + p h (b ) F (x) . k j j+1 j k j,k j+1,k Using (3.3) for Fj (x) gives the relation Fj (x) =

1 pj qj bTj Fj−1 (x) − qj gjT rj−1 (x) pj+1 qj

(3.8)

Inserting (3.8) into (3.7) gives 1 ' (x − dk )rk−1 (x) − gk−1 hk rk−2 (x) − · · · rk (x) = pk+1 qk

( T T × · · · − g j b× h r (x) + p h (b ) F (x) . j k j−1,k j−1 j,k k j−1

Therefore since (3.6) is the case of (3.7) for j = k − 2, (3.7) is true for each j = k − 2, k − 3, . . . , 0, and for j = 0, using the fact that F0 = 0 we have ( ' 1 rk (x) = (x − dk )rk−1 (x) − gk−1 hk rk−2 (x) − · · · − g1 b× 1,k hk r0 (x) pk+1 qk Since these coeﬃcients coincide with (3.2) that are satisﬁed by the polynomial system R, the polynomials given by (3.3) must coincide with these polynomials. This proves the theorem.

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4. Recurrence relations for polynomials associated with (H, m)-quasiseparable polynomials In the previous section, sparse recurrence relations for (H, m)-quasiseparable polynomials were derived. However, the classical Traub algorithm is based on the recurrence relations for the Horner polynomials, not the original monomial system. In this section, sparse recurrence relations for the system of polynomials associated with a system of (H, m)-quasiseparable polynomials (i.e., the generalized Horner polynomials) are derived. It is these recurrence relations that form the basis of the Traub-like algorithm. 4.1. Introduction of a perturbation term via pertransposition of confederate matrices Let R = {rk (x)}nk=0 be a system of (H, m)-quasiseparable polynomials, and {xk }nk=1 a set of distinct nodes. Decomposing the master polynomial into the R basis as n

(x − xk ) =: P (x) = P0 · r0 (x) + P1 · r1 (x) + · · · + Pn−1 · rn−1 (x) + Pn · rn (x)

k=1

yields the coeﬃcients P0 , P1 , . . . , Pn , and we have @ d @ 1@ gi b× ij hj @ @ . .@ p2 q1@ . @ @ CR (P ) = @ @ @ − P1n .. @ @ @ . @ @ .@ .. @ @ @ @ @ @ 0 q p @ n n−1@ dn@ @ @

P0

.. .

0

Pn−1

(4.1)

Applying (2.8) gives us the confederate matrix for the associated polynomials as @ d @ n @ gn−j b× n−j,n−i hn−i @ @ . .@ @ . pn qn−1@ @ CR) (P ) = @ @ @ − P1n .. @ @ @ . @ @ .@ .. @ @ @ @ @ @ 0 p q @ 2 1 @ d1@ @ @

Pn−1

···

P0

0

(4.2)

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From this last equation we can see that the n-term recurrence relations satisﬁed ) are given by by the associated polynomials R r)k (x) =

1 p)k+1 q)k

rk−1 (x) − (x − d)k ))

) r ) (x) − g)j+1)b× h k j j+1,k

k−2 j=0

typical term as in (3.2)

Pn−k r)0 (x) P n perturbation term

(4.3) where, in order to simplify the formulas, we introduce the notation p)k = qn−k+1 ,

q)k = pn−k+1 ,

d)k = dn−k+1 ,

g)k = hTn−k+1 ,

)bk = bT n−k+1 ,

T ) hk = gn−k+1 .

(4.4)

We will see in a moment that the nonzero top row of the second matrix in (4.2) introduces perturbation terms into all formulas that we derive. These perturbation terms are the cause of the increase in quasiseparable rank by at most one. 4.2. Perturbed [EGO05]-type recurrence relations The previous section showed how n-term recurrence relations for a system of (H, m)-quasiseparable polynomials change after a rank one perturbation of the ﬁrst row of the corresponding confederate matrix. This small increase in quasiseparable rank allows construction of the desired sparse recurrence relations for associated polynomials, as presented in the next theorem. Theorem 4.1 (Perturbed [EGO05]-type recurrence relations). Let R = {r0 (x), . . . , rn−1 (x), P (x)} be a system of (H, m)-quasiseparable polynomials corresponding to a (H, m)-quasiseparable matrix of size n×n with generators {pk , qk , dk , gk , bk , hk } as in Deﬁnition ) 3.1, with the convention that qn = 0, bn = 0. Then the system of polynomials R associated with R satisfy the recurrence relations ⎤

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

F0 (x)

r0 (x)

⎡

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣

⎤ 0

Pn

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(4.5)

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⎡

⎤

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ 1 ⎥= ⎥ p ) )k k+1 q ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ p) q) ⎢ k k ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ p)k

F)k (x)

r)k (x)

)bT k

) hTk

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ 1 ⎢ + p)k+1 q)k ⎢ ⎢ ⎢ ⎢ ⎣

⎤ 0

Pn−k

−) qk g)kT

x − d)k

⎤⎡

⎤

⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

F)k−1 (x)

r)k−1 (x)

typical terms

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(4.6)

perturbation term

with the vector of auxiliary polynomials F)k (x), and the coeﬃcients Pk , k = 0, . . . , n are as deﬁned in (1.3). Proof. The recurrence relations (4.6) deﬁne a system of polynomials which satisfy the n-term recurrence relations r)k (x) = (αk x − ak−1,k ) · r)k−1 (x) − ak−2,k · r)k−2 (x) − · · · − a0,k · r)0 (x)

(4.7)

for some coeﬃcients αk , ak−1,k , . . . , a0,k . The proof is presented by showing that these n-term recurrence relations in fact coincide exactly with (4.3), so these coeﬃcients coincide with those of the n-term recurrence relations of the associated ) that is, polynomials R;

1 1 ) 1 Pn−k ) αk = , ak−1,k = − g1)b× ) dk , a0,k = h 1,k k p)k+1 q)k p)k+1 q)k p)k+1 q)k Pn (4.8) 1 ) h aj,k = g)j+1)b× , j = 1, . . . , k − 2 k j+1,k p)k+1 q)k Using relations for r)k (x) and F)k−1 (x) from (4.6), we have

1 rk−1 (x) − g)k−1 ) r)k (x) = (x − d)k )) hk r)k−2 (x) p)k+1 q)k Pn−k T )T ) ) r)0 (x) . + p)k−1 hk bk−1 Fk−2 (x) + Pn

(4.9)

Notice that again using (4.6) to eliminate F)k−2 (x) from the equation (4.9) will result in an expression for r)k (x) in terms of r)k−1 (x), r)k−2 (x), r)k−3 (x), F)k−3 (x),

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and r)0 (x) without modifying the coeﬃcients of r)k−1 (x), r)k−2 (x), or r)0 (x). Again applying (4.6) to eliminate F)k−3 (x) results in an expression in terms of r)k−1 (x), r)k−2 (x), r)k−3 (x), r)k−4 (x), F)k−4 (x), and r)0 (x) without modifying the coeﬃcients of r)k−1 (x), r)k−2 (x), r)k−3 (x), or r)0 (x). Continuing in this way, the n-term recurrence relations of the form (4.7) are obtained without modifying the coeﬃcients of the previous ones. Suppose that for some 0 < j < k − 1 the expression for r)k (x) is of the form

1 rk−1 (x) − g)k−1 ) (4.10) r)k (x) = (x − d)k )) hk r)k−2 (x) − · · · p)k+1 q)k Pn−k × T )× T ) ) ) ) · · · − g)j+1 bj+1,k hk r)j (x) + p)j+1 hk (bj,k ) Fj (x) + r)0 (x) . Pn Using (4.6) for F)j (x) gives the relation 1 gjT r)j−1 (x) F)j (x) = p)j q)j )bTj F)j−1 (x) − q)j ) p)j+1 q)j

(4.11)

Inserting (4.11) into (4.10) gives

1 rk−1 (x) − ) gk−1 ) (x − d)k )) hk r)k−2 (x) − · · · r)k (x) = p)k+1 q)k

(4.12) T Pn−k T )× ) ) ) ··· − ) gj)b× r ) (x) + p ) ( b ) (x) + r ) (x) . F h h j k j−1,k j−1 0 j,k k j−1 Pn

Therefore since (4.9) is the case of (4.10) for j = k − 2, (4.10) is true for each j = k − 2, k − 3, . . . , 0, and for j = 0, using the fact that F)0 = 0 we have

1 r)k (x) = rk−1 (x) − g)k−1 ) (x − d)k )) hk r)k−2 (x) − · · · p)k+1 q)k (4.13) Pn−k × ) ) · · · − g)1 b1,k hk r)0 (x) + r)0 (x) Pn Since these coeﬃcients coincide with those in (4.8) that are satisﬁed by the associated polynomials, the polynomials given by (4.6) must coincide with the associated polynomials. This proves the theorem. 4.3. Known special cases of these more general recurrence relations In this section, recurrence relations valid for the class of polynomials associated with (H, m)-quasiseparable polynomials were derived. Such are needed to provide a Traub-like algorithm. We next demonstrate that these more general recurrence relations reduce as expected in the classical cases. That is, since monomials and real orthogonal polynomials are themselves (H, m)-quasiseparable, the above formulas are valid for those classes as well, and furthermore, the special cases of these formulas are, in fact, the classical formulas for these cases. Example 4.2 (Classical Traub case: monomials and the Horner polynomials). As shown earlier, the well-known companion matrix (2.5) results when the polynomial

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system R is simply a system of monomials. By choosing the generators pk = 1, qk = 1, dk = 0, gk = 1, bk = 1, and hk = 0, the matrix (4.1) reduces to (2.5), and also (4.2) reduces to the confederate matrix for the Horner polynomials (2.7). In this special case, the perturbed three-term recurrence relations of Theorem 4.1 become r)0 (x) = Pn ,

r)k (x) = x) rk−1 (x) + Pn−k ,

(4.14)

after eliminating the auxiliary polynomials present, coinciding with the known recurrence relations for the Horner polynomials, used in the evaluation of the polynomial (4.15) P (x) = P0 + P1 x + · · · + Pn−1 xn−1 + Pn xn . Example 4.3 (Calvetti–Reichel case: Real orthogonal polynomials and the Clenshaw rule). Consider the almost tridiagonal confederate matrix ⎤ ⎡ d1 h2 0 · · · 0 −P0 /Pn .. ⎥ ⎢ .. ⎥ ⎢ q1 d2 h3 . . −P1 /Pn ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ 0 q2 d3 h4 0 . ⎥. ⎢ (4.16) CR (P ) = ⎢ ⎥ . . ⎥ ⎢ 0 . q d /P 0 −P 3 4 n−3 n ⎥ ⎢ ⎥ ⎢ . . .. .. ... ... ⎣ .. . h −P /P ⎦ 0

···

n

0

0

qn−1

n−2

n

dn − Pn−1 /Pn

The corresponding system of polynomials R satisfy three-term recurrence relations; for instance, the highlighted column implies r3 (x) =

1 h3 (x − d3 )r2 (x) − r1 (x) q3 q3

(4.17)

by the deﬁnition of the confederate matrix. Thus, confederate matrices of this form correspond to systems of polynomials satisfying three-term recurrence relations, or systems of polynomials orthogonal on a real interval, and the polynomial P (x). Such confederate matrices can be seen as special cases of our general class by choosing scalar generators, with pk = 1, bk = 0, and gk = 1, and in this case the matrix (4.1) reduces to (4.16). With these choices of generators, applying Theorem 4.1 and eliminating the auxiliary polynomials yields the recurrence relations r)k (x) =

1 qn−k+1 1 (x − dn−k )) rk−1 (x) − hn−k+1 r)k−2 (x) + Pn−k qn−k qn−k qn−k

(4.18)

which coincides with the Clenshaw rule for evaluating a polynomial in a real orthogonal basis, i.e., of the form P (x) = P0 r0 (x) + P1 r1 (x) + · · · + Pn−1 rn−1 (x) + Pn rn (x).

(4.19)

By the discussion of pertransposition in Section 2, recurrence relations for the system of polynomials associated with real orthogonal polynomials can be found

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by considering the confederate matrix ⎡ ⎤ dn − Pn−1 /Pn hn − Pn−2 /Pn −Pn−3 /Pn · · · −P1 /Pn −P0 /Pn .. ⎢ ⎥ .. ⎢ ⎥ . . qn−1 dn−1 hn−1 ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ 0 qn−2 dn−2 hn−2 0 . ⎥. CR (P ) = ⎢ ⎢ ⎥ .. ⎢ ⎥ . qn−3 dn−3 0 0 ⎢ ⎥ ⎢ ⎥ . . . . . .. .. .. .. .. ⎣ h2 ⎦ 0 ··· 0 0 q1 d1 (4.20) obtained by pertransposition of (4.16). Note that the highlighted column corresponds to the full recurrence relation 1 hn−1 1 Pn−3 r)3 (x) = (x − dn−2 )) r2 (x) − r)1 (x) + r)0 (x) (4.21) qn−3 qn−3 qn−3 Pn Thus our formula generalizes both the Clenshaw rule and the algorithms designed for inversion of three-term-Vandermonde matrices in [7] and [14].

5. Computing the coeﬃcients of the master polynomial Note that in order to use the recurrence relations of the previous section it is necessary to decompose the master polynomial P (x) into the R basis; that is, the coeﬃcients Pk as in (1.3) must be computed. To this end, an eﬃcient method of calculating these coeﬃcients follows. It is easily seen that the last polynomial rn (x) in the system R does not aﬀect the resulting confederate matrix CR (P ). Thus, if ¯ = {r0 (x), . . . , rn−1 (x), xrn−1 (x)}, R ¯ basis can we have CR (P ) = CR¯ (P ). Decomposing the polynomial P (x) into the R (0) be done recursively by setting rn (x) = 1 and then for k = 0, . . . , n − 1 updating (k+1) (k) rn (x) = (x − xk+1 ) · rn (x). The following lemma gives this procedure, and is from [23]. Lemma 5.1 ([23]). Let R = {r0 (x), . . . , rn (x)} be given by (1.2), and f (x) = k+1 k i=1 ai · ri (x), where k < n − 1. Then the coeﬃcients of x · f (x) = i=1 bi · ri (x) can be computed by ⎤ ⎡ ⎡ ⎤ a0 b0 ⎢ .. ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ bk ⎥

⎢ ak ⎥ ⎥ ⎢ ⎢ ⎥ C (r ) 0 R n ⎢ 0 ⎥ ⎢ bk+1 ⎥ = (5.1) 1 ⎥ ⎢ ⎢ ⎥ 0 · · · 0 0 αn ⎢ 0 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎣ .. ⎦ ⎣ .. ⎦ 0 0

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Proof. It can be easily checked that x · r0 (x) r1 (x) · · · rn (x) − =

0

· · · rn (x) · · · 0 x · rn (x) . r0 (x)

r1 (x)

·

CR (rn ) 0 ··· 0

1 αn

0 0

Multiplying the latter equation by the column of the coeﬃcients we obtain (5.1). This lemma suggests the following algorithm for computing coeﬃcients {P0 , P1 , . . . , Pn−1 , Pn } in n

(x − xk ) = P0 r0 (x) + P1 r1 (x) + · · · + Pn−1 rn−1 (x) + Pn rn (x).

(5.2)

k=1

of the master polynomial. Algorithm 5.2 (Coeﬃcients of the master polynomial in the R basis). Cost: O(n × m(n)), where m(n) is the cost of multiplication of an n × n quasiseparable matrix by a vector. Input: A quasiseparable confederate matrix CR (rn ) and n nodes x = (x1 ,x2 ,...,xn ). ' ( (0) (0) 1. Set P0(0) · · · Pn−1 = 1 0 ··· 0 Pn 2. For k = 1 : n, ⎡ (k−1) ⎤ ⎡ (k) ⎤ P0 P0 ⎢ . ⎥

⎥

⎢ .. ⎢ ⎢ .. ⎥ ⎥ CR¯ (x · rn−1 (x)) 0 . ⎢ ⎢ ⎥= ⎥ − x · I · k ⎢ ⎢ (k) ⎥ ⎥ 0 ··· 0 1 0 ⎣ P (k−1) ⎦ ⎣ P ⎦ n−1 (k)

n−1 (k−1)

Pn

¯ = {r0 (x), . . . , rn−1 (x), xrn−1 (x)}. where R ' (n) 3. Take P0 · · · Pn−1 Pn = P0(n) · · · Pn−1

Pn

(n)

(

Pn

Output: Coeﬃcients {P0 , P1 , . . . , Pn−1 , Pn } such that (5.2) is satisﬁed. It is clear that the computational burden in implementing this algorithm is in multiplication of the matrix CR¯ (rn ) by the vector of coeﬃcients. The cost of each such step is O(m(n)), where m(n) is the cost of multiplication of an n × n quasiseparable matrix by a vector, thus the cost of computing the n coeﬃcients is O(n × m(n)). Using a fast O(n) algorithm for multiplication of a quasiseparable matrix by a vector ﬁrst derived in [9] (or its matrix interpretation of [4]), the cost of this algorithm is O(n2 ).

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6. The overall Traub-like algorithm 6.1. Quasiseparable generator input The main algorithm of this section is the Traub-like algorithm that outputs the inverse of a (H, m)-quasiseparable-Vandermonde matrix. It takes as input the generators {pk , qk , dk , gk , bk , hk } of the (H, m)-quasiseparable confederate matrix corresponding to the system of polynomials R. In this algorithm we will make use of MATLAB notations; for instance VR) (i : j, k : l) will refer to the block of VR) (x) consisting of rows i through j and columns k through l. For each node xk we have a vector of auxiliary polynomials FR) (xk ). *k Let us compose a matrix of such vectors [FR) (x1 )| · · · |FR) (xn )] and denote it as F on each step. Algorithm 6.1 (Traub-like inversion algorithm). Cost: O(n2 ) operations. Input: Generators {pk , qk , dk , gk , bk , hk } of a quasiseparable confederate matrix corresponding to a system of polynomials R and n nodes x = (x1 , x2 , . . . , xn ). n 1. Compute the entries of diag(c1 , . . . , cn ) via (2.2): ci = (xk − xi )−1 . k=1

k =i

2. Compute the coeﬃcients {P0 , . . . , Pn } of the master polynomial P (x) as in (1.3) via Algorithm 5.2. ) speciﬁed via (2.8) at the n nodes xk to form 3. Evaluate the n polynomials of R VR) (x). Theorems 4.1 provides an algorithm for this. ⎤ ⎡ ⎡ ⎤ 1 0 ··· 0 ⎢ 1 ⎥ ⎥ ⎢ . . ⎥ *1 = ⎢ F (a) Set VR) (:, 1) = Pn ⎢ . ⎥, ⎣ .. . . . .. ⎦. ⎣ .. ⎦ 0 ··· 0 1 (b) For k = 1 : n − 1, compute ⎛⎡ ⎤ ⎛ ⎡ ⎤⎞ ⎡ ⎤⎞ x1 1 1 ⎜⎢ x2 ⎥ ⎜ ⎢ 1 ⎥⎟ ⎢ 1 ⎥⎟ 1 ⎜⎢ ⎥ ⎜ *) ⎢ ⎥⎟ ⎢ ⎥⎟ VR) (:, k + 1) = ⎜p)k Fk hk + ⎜⎢ .. ⎥ − d)k ⎢ .. ⎥⎟ VR) (:, k) + Pn−k ⎢ .. ⎥⎟ ⎝⎣ . ⎦ ⎣ . ⎦⎠ ⎣ . ⎦⎠ p)k+1 q)k ⎝ xn and FR) (:, k + 1) =

1

1

1

*k − q)k g)kT V ) (:, k)T p)k q)k)bTk F R

p)k+1 q)k Note: The product of two column vectors is understood to be componentwise. 4. Compute the inverse VR (x)−1 via (2.1): VR (x)−1 = I˜ · V T (x) · diag(c1 , . . . , cn ) ) R

−1

Output: Entries of VR (x)

, the inverse of the polynomial-Vandermonde matrix.

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6.2. Recurrence relation coeﬃcient input The previous section provides the Traub-like algorithm, which takes as input the generators of the (H, m)-quasiseparable polynomials involved in forming the quasiseparable-Vandermonde matrix. However, as in the motivating cases of real orthogonal polynomials and Szeg¨ o polynomials, problems may be stated in terms of the coeﬃcients of the involved recurrence relations instead of in terms of generators. In this section, we present a result allowing conversion from the language of recurrence relation coeﬃcients to that of quasiseparable generators. Applying this conversion as a preprocessor, the algorithm of the previous section can then be used for problems stated in terms of recurrence relation coeﬃcients. Theorem 6.2 (Recurrence relations coeﬃcients ⇒ quasiseparable generators). Let R = {rk (x)}nk=0 be a system of polynomials satisfying the [EGO05]-type two-term recurrence relations (3.4):

αk βk Fk−1 (x) Fk (x) = . rk (x) rk−1 (x) γk δk x + θk Then the (H, m)-quasiseparable matrix CR (rn ) = ⎡ θ1 − δ1 −( δ12 )γ2 β1 − δ13 γ3 α2 β1 − δ14 γ4 α3 α2 β1 ⎢ 1 − θδ22 − δ13 γ3 β2 − δ14 γ4 α3 β2 ⎢ δ1 ⎢ ⎢ 0 1 − θδ33 − δ14 γ4 β3 ⎢ δ2 ⎢ ⎢ 1 0 − θδ44 ⎢ 0 δ3 ⎢ ⎢ .. .. .. .. . . . ⎣ . 0 ··· 0 0

··· ··· .. . .. . .. .

(6.1) ⎤

− δ1n γn αn−1 αn−2 · · · α3 α2 β1 − δ1n γn αn−1 αn−2 · · · α3 β2 − δ1n γn αn−1 · · · α4 β3 .. .

1 δn−1

− δ1n γn βn−1 − θδnn

with generators dk = −(θk /δk ),

(k = 1, . . . , n),

pk+1 qk = (1/δk ),

(k = 1, . . . , n − 1),

gk

=

βkT

,

(k = 1, . . . , n − 1),

bk

=

αTk

,

(k = 2, . . . , n − 1),

hk = −

1 T γ , δk k

(k = 2, . . . , n)

is a confederate matrix for the system of polynomials R.

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

A Traub-like Algorithm

149

Proof. Inserting the speciﬁed choice of generators into the general n-term recurrence relations (3.2), we arrive at rk (x) = (δk x + θk )rk−1 (x) + γk βk−1 rk−2 (x) + γk αk−1 βk−2 rk−3 (x) + γk αk−1 αk−2 βk−3 rk−4 (x) + · · · + γk αk−1 · · · α2 β1 r0 (x)

(6.2)

It suﬃces to show that the polynomial system satisfying the two-term recurrence relations also satisﬁes these n-term recurrence relations. Beginning with rk (x) = (δk x + θk )rk−1 (x) + γk Fk−1 (x)

(6.3)

and using the relation Fk−1 (x) = αk−1 Fk−2 (x) + βk−1 rk−2 (x), (6.3) becomes rk (x) = (δk x + θk )rk−1 (x) + γk βk−1 rk−2 (x) + γk αk−1 Fk−2 (x) and continuing this procedure to obtain n-term recurrence relations. It can easily be checked that this procedure yields exactly (6.2).

7. Numerical Experiments The numerical properties of the Traub algorithm and its generalizations (that are the special cases of the algorithm proposed in this paper) were studied by diﬀerent authors. It was noticed in [16] that a modiﬁcation of the classical Traub algorithm of [28] can yield high accuracy in certain cases if the algorithm is preceded with the Leja ordering of the nodes; that is, ordering such that |x1 | = max |xi |, 1in

k−1 j=1

|xk − xj | = max

kin

k−1

|xi − xj |,

k = 2, . . . , n − 1

j=1

(see [26], [19], [24]) It was noticed in [16] that the same is true for ChebyshevVandermonde matrices. No error analysis was done, but the conclusions of the above authors was that in many cases the Traub algorithm and its extensions can yield much better accuracy than Gaussian elimination, even for very ill-conditioned matrices. We made our preliminary experiments with the proposed Traub-like algorithm, and our conclusions for the most general case are consistent with the experience of our colleagues made for special cases. The results of experiments with the proposed algorithm yields better accuracy than Gaussian elimination. However, our experiments need to be done for diﬀerent special cases of (H, m)quasiseparable polynomials. The algorithm was implemented in C using Lapack for all supplementary matrix computations (such as matrix multiplication GEMM). For Gaussian elimination we used the Lapack subroutine GESV. To estimate the accuracy of all of the above algorithms we took the output of new Traub-like algorithm in double precision VR (x)−1 as the exact solution.

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We compare the forward accuracy of the inverse computed by the algorithm in single precision VR (x)−1 with respect to the inverse computed in double precision, deﬁned by (x)−1 − VR (x)−1 2 VR (7.1) e= VR (x)−1 2 where VRs (x)−1 is the solution computed by each algorithm in single precision. In the tables, New Algorithm denotes the proposed Traub-like algorithm with Leja ordering, and GESV indicates Lapack’s inversion subroutine. Finally, cond(V ) denotes the condition number of the matrix V computed via the Lapack subroutine GESVD. Experiment 1. (Random choice of generators) In this experiment, the generators we chosen randomly in (−1, 1), and the nodes xk were selected equidistant on (−1, 1) via the formula

k , k = 0, 1, . . . , n − 1 xk = −1 + 2 n−1 We test the accuracy of the inversion algorithm for various sizes n and quasiseparable ranks m of matrices generated in this way. Some results are tabulated in Table 4. Notice that the performance of the proposed inversion algorithm is an improvement over that of Lapack’s standard inversion subroutine GESV in this speciﬁc case. And in almost all cases relative errors are around e-7, which means that all digits of the errors in single precision coincide with corresponding digits in double precision. There are occasional examples in which the proposed algorithm can lose several decimal digits, but it still outperforms Gaussian elimination. Experiment 2. (l-recurrent polynomials) In this experiment we consider l-recurrent polynomials rk (x) = (αk x−ak−1,k )·rk−1 (x)−ak−2,k ·rk−2 (x)−· · ·−ak−(l−1),k ·rk−(l−1) (x) (7.2) by choosing coeﬃcients of (7.2) randomly in (−1, 1), and the nodes xk equidistant on (−1, 1). We test the accuracy of the inversion algorithm for various sizes n and number of terms l. We remind the reader that quasiseparable rank of polynomials given by (7.2) is l − 2. Table 4 presents some results generated in this way.

8. Conclusions In this paper we extend the previous work in the area of fast Traub-like inversion algorithms to the general class of (H, m)-quasiseparable-Vandermonde matrices. This generalizes results for Vandermonde, three-term-Vandermonde, Szeg¨o– Vandermonde, and (H, 1)-quasiseparable-Vandermonde matrices. Exploiting the quasiseparable structure yields sparse recurrence relations which allow the desired computational speedup, resulting in a fast O(n2 ) algorithm as opposed to Gaussian

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151

Table 4. Random generators on (−1, 1). Equidistant nodes on (−1, 1). n

m

cond(V )

GESV

New Algorithm

10

1 2 1 2 3 4 1 2 3 4 5 6 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

1.8e+007 5.6e+007 2.2e+020 1.6e+019 1.0e+021 6.1e+020 7.2e+029 3.4e+025 2.9e+029 7.5e+026 5.0e+024 2.5e+026 2.1e+034 3.3e+033 4.1e+029 4.5e+028 1.2e+031 3.5e+032 6.0e+027 7.8e+031 1.5e+039 3.7e+038 2.6e+041 2.0e+037 1.7e+037 8.0e+038 1.7e+038 7.5e+036

1.5e-006 4.6e-005 5.9e-001 2.6e+000 5.6e-002 2.8e+000 1.2e+000 9.2e-001 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000

1.4e-006 5.4e-007 5.0e-007 1.9e-007 2.3e-006 5.5e-006 2.6e-006 2.7e-006 2.0e-006 1.5e-006 1.4e-006 5.1e-007 1.6e-005 1.9e-006 1.1e-003 3.5e-007 7.2e-007 3.3e-006 1.7e-004 5.0e-007 3.9e-007 7.6e-001 4.5e-004 3.8e-001 5.7e-007 9.2e-005 9.3e-007 4.7e-007

20

30

40

50

elimination, which requires O(n3 ) operations. Finally, some numerical experiments were presented that indicate that, under some circumstances, the resulting algorithm can give better performance than Gaussian elimination.

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l

cond(V )

inv()

TraubQS(Leja)

10

3 4 3 4 5 6 3 4 5 6 7 8 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10 11 12

9.5e+004 1.2e+006 4.3e+013 2.2e+013 1.5e+012 4.1e+011 4.7e+016 1.8e+016 3.0e+018 7.3e+016 1.2e+017 3.6e+017 8.9e+017 1.2e+020 2.3e+018 2.2e+021 2.4e+020 1.8e+018 8.9e+019 8.6e+020 3.6e+018 3.1e+019 5.3e+019 7.8e+019 1.8e+020 3.6e+019 7.0e+019 2.2e+020 3.9e+021 5.3e+020

1.5e-004 7.1e-004 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000 1.0e+000

1.9e-007 3.3e-007 2.8e-007 3.4e-007 5.1e-007 2.2e-007 4.2e-007 3.2e-007 4.4e-007 4.1e-007 5.1e-007 2.8e-007 4.8e-007 6.5e-007 8.3e-007 4.5e-007 6.8e-007 9.7e-007 1.1e-006 6.3e-007 2.8e-007 2.0e-007 6.1e-007 4.8e-007 1.8e-007 4.2e-007 5.5e-007 8.5e-007 1.8e-007 5.3e-007

20

30

40

50

References [1] M.Bakonyi and T.Constantinescu, Schur’s algorithm and several applications, in Pitman Research Notes in Mathematics Series, vol. 61, Longman Scientiﬁc and Technical, Harlow, 1992. [2] A. Bj¨ orck and V. Pereyra, Solution of Vandermonde Systems of Equations, Math. Comp., 24 (1970), 893–903.

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[3] T. Bella, Y. Eidelman, I. Gohberg, I. Koltracht and V. Olshevsky, A Bj¨ orck–Pereyratype algorithm for Szeg¨ o-Vandermonde matrices based on properties of unitary Hessenberg matrices, Linear Algebra and Applications, Volume 420, Issues 2-3 pp. 634– 647, 2007. [4] T. Bella, Y. Eidelman, I. Gohberg, I. Koltracht and V. Olshevsky A fast Bjorck– Pereyra like algorithm for solving Hessenberg-quasiseparable-Vandermonde systems, submitted to SIAM Journal of Matrix Analysis (SIMAX), 2007. [5] T. Bella, Y. Eidelman, I. Gohberg, V. Olshevsky, Classiﬁcations of three-term and two-term recurrence relations and digital ﬁlter structures via subclasses of quasiseparable matrices, submitted to SIAM Journal of Matrix Analysis (SIMAX), 2007. [6] T. Bella, Y. Eidelman, I. Gohberg, V. Olshevsky, E. Tyrtyshnikov Fast Traub-like inversion algorithm for Hessenberg order one quasiseparable Vandermonde matrices, submitted to Journal of Complexity, 2007. [7] Calvetti, D. and Reichel, L., Fast inversion of Vandermonde-like matrices involving orthogonal polynomials, BIT, 1993. [8] Y. Eidelman and I. Gohberg, On a new class of structured matrices, Integral Equations and Operator Theory, 34 (1999), 293–324. [9] Y. Eidelman and I. Gohberg, Linear complexity inversion algorithms for a class of structured matrices, Integral Equations and Operator Theory, 35 (1999), 28–52. [10] Y. Eidelman and I. Gohberg, A modiﬁcation of the Dewilde-van der Veen method for inversion of ﬁnite-structured matrices, Linear Algebra Appl., 343–344 (2002), 419–450. [11] Y. Eidelman, I. Gohberg and V. Olshevsky, Eigenstructure of Order-One-Quasiseparable Matrices. Three-term and Two-term Recurrence Relations, Linear Algebra and its Applications, Volume 405, 1 August 2005, Pages 1–40. [12] G. Forney, Concatenated codes, The M.I.T. Press, 1966, Cambridge. [13] L.Y. Geronimus, Polynomials orthogonal on a circle and their applications, Amer. Math. Translations, 3 p. 1–78, 1954 (Russian original 1948). [14] I. Gohberg and V. Olshevsky, Fast inversion of Chebyshev-Vandermonde matrices, Numerische Mathematik, 67, No. 1 (1994), 71–92. [15] I. Gohberg and V. Olshevsky, A fast generalized Parker-Traub algorithm for inversion of Vandermonde and related matrices, Journal of Complexity, 13(2) (1997), 208–234. A short version in Proceedings in Communications, Computation, Control and Signal Processing: A tribute to Thomas Kailath, Eds. A. Paulraj, V. Roychowdhury and C. Shaper, Kluwer Academic Publishing, 1996, 205–221. [16] I. Gohberg and V. Olshevsky, The fast generalized Parker-Traub algorithm for inversion of Vandermonde and related matrices, J. of Complexity, 13(2) (1997), 208–234. [17] U. Grenader and G. Szeg¨ o, Toeplitz forms and Applications, University of California Press, 1958. [18] W.G. Horner, A new method of solving numerical equations of all orders by continuous approximation, Philos. Trans. Roy. Soc. London, (1819), 308–335. [19] N.J. Higham, Stability analysis of algorithms for solving conﬂuent Vandermonde-like systems, SIAM J. Matrix Anal. Appl., 11(1) (1990), 23–41.

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[20] T. Kailath and V. Olshevsky, Displacement structure approach to polynomial Vandermonde and related matrices, Linear Algebra and Its Applications, 261 (1997), 49–90. [21] J. Maroulas and S. Barnett, Polynomials with respect to a general basis. I. Theory, J. of Math. Analysis and Appl., 72 (1979), 177–194. [22] V. Olshevsky, Eigenvector computation for almost unitary Hessenberg matrices and inversion of Szeg¨ o-Vandermonde matrices via Discrete Transmission lines. Linear Algebra and Its Applications, 285 (1998), 37–67. [23] V. Olshevsky, Associated polynomials, unitary Hessenberg matrices and fast generalized Parker–Traub and Bjorck–Pereyra algorithms for Szeg¨o-Vandermonde matrices invited chapter in the book “Structured Matrices: Recent Developments in Theory and Computation,” 67–78, (D. Bini, E. Tyrtyshnikov, P. Yalamov., Eds.), 2001, NOVA Science Publ., USA. [24] V. Olshevsky, Pivoting for structured matrices and rational tangential interpolation, in Fast Algorithms for Structured Matrices: Theory and Applications, CONM/323, 1–75, AMS publications, May 2003. [25] F. Parker, Inverses of Vandermonde matrices, Amer. Math. Monthly, 71 (1964), 410– 411. [26] L. Reichel and G. Opfer, Chebyshev-Vandermonde systems, Math. of Comp., 57 (1991), 703–721. [27] J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, 1992. 277–301. [28] J. Traub, Associated polynomials and uniform methods for the solution of linear problems, SIAM Review, 8, No. 3 (1966), 277–301. T. Bella, V. Olshevsky and P. Zhlobich Department of Mathematics University of Connecticut Storrs CT 06269-3009, USA e-mail: [email protected] (contact author) [email protected] Y. Eidelman and I. Gohberg School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University Ramat-Aviv 69978, Israel e-mail: [email protected] [email protected] E. Tyrtyshnikov Institute of Numerical Mathematics Russian Academy of Sciences Gubkina Street, 8 Moscow, 119991, Russia e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 199, 155–173 c 2010 Birkh¨ auser Verlag Basel/Switzerland

A Fast Algorithm for Approximate Polynomial GCD Based on Structured Matrix Computations Dario A. Bini and Paola Boito In Memory of Georg Heinig

Abstract. An O(n2 ) complexity algorithm for computing an -greatest common divisor (gcd) of two polynomials of degree at most n is presented. The algorithm is based on the formulation of polynomial gcd given in terms of resultant (B´ezout, Sylvester) matrices, on their displacement structure and on the reduction of displacement structured matrices to Cauchy-like form originally pointed out by Georg Heinig. A Matlab implementation is provided. Numerical experiments performed with a wide variety of test problems, show the eﬀectiveness of this algorithm in terms of speed, stability and robustness, together with its better reliability with respect to the available software. Mathematics Subject Classiﬁcation (2000). 68W30, 65F05, 15A23. Keywords. Cauchy matrices, polynomial gcd, displacement structure, Sylvester matrix, B´ezout matrix.

1. Introduction A basic problem in algebraic computing is the evaluation of polynomial gcd: given the coeﬃcients of two univariate polynomials n m ui xi , v(x) = vi xi , u(x) = i=0

i=0

compute the coeﬃcients of their greatest common divisor g(x). In many applications, input data are represented as ﬂoating point numbers or derive from the results of physical experiments or previous computations, so Supported by MIUR grant 2006017542.

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that they are generally aﬀected by errors. If u(x) and v(x) have a nontrivial gcd, it turns out that arbitrarily small perturbations in the coeﬃcients of u(x) and v(x) may transform u(x) and v(x) into relatively prime polynomials. Therefore, it is clear that the concept of gcd is not well suited to deal with applications where data are approximatively known. This is why the notion of approximate gcd, or -gcd, has been introduced. For more details on this topic we refer the reader to [19] [7],[18], [23] and to the references therein. We use the following deﬁnition where · denotes the Euclidean norm. Deﬁnition 1.1. A polynomial g(x) is said to be an -divisor of u(x) and v(x) if there exist polynomials u ˆ(x) and vˆ(x) of degree n and m, respectively, such that u(x) − u ˆ(x) ≤ u(x), v(x) − vˆ(x) ≤ v(x) and g(x) divides u ˆ(x) and vˆ(x). If g(x) is an -divisor of maximum degree of u(x) and v(x), then it is called -gcd of u(x) and v(x). The polynomials p(x) = u ˆ(x)/g(x) and q(x) = vˆ(x)/g(x) are called -cofactors. Several algorithms for the computation of an approximate polynomial gcd can be found in the literature; they rely on diﬀerent techniques, such as the Euclidean algorithm [1], [2], [12], [17], optimization methods [15], SVD and factorization of resultant matrices [5], [4], [23], Pad´e approximation [3], [18], root grouping [18]. Some of them have been implemented inside numerical/symbolic packages like the algorithm of Zeng [23] in MatlabTM and the algorithms of Kaltofen [14], of Corless et al. [4], of Labahn and Beckermann [13] in MapleTM . These algorithms have a computational cost of O(n3 ) which makes them expensive for moderately large values of n. Algorithms based on the Euclidean scheme have a typical cost of O(n2 ) but they are prone to numerical instabilities; look-ahead strategies can improve the numerical stability with an increase of the complexity to O(n3 ). More recently, O(n2 ) algorithms have been proposed in [24] and [16]. They are based on the QR factorization of a displacement structured matrix obtained by means of the normal equations. The use of the normal equations generally squares the condition number of the original problem with a consequent deterioration of the stability. In this paper we present an algorithm for (approximate) gcd computation which has a cost of O(n2 ) arithmetic operations and, from the several numerical experiments performed so far, results robust and numerically stable. The algorithm relies on the formulation of the gcd problem given in terms of the B´ezout matrix B(u, v) or of the Sylvester matrix S(u, v) associated with the pair of polynomials (u, v), and on their reduction to Cauchy-like matrices by means of unitary transforms. This kind of reduction, which is fundamental for our algorithm, was discovered and analyzed by Georg Heinig in [10] in the case of general Toeplitz-like matrices. For exact gcd, where = 0, the degree k of the -gcd coincides with the nullity (i.e., the dimension of the kernel) of B(u, v) and of S(u, v), or equivalently, with the nullity of the Cauchy matrices obtained through the reduction.

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Our algorithm can be divided into two stages. In the ﬁrst stage, from the coeﬃcients of the input polynomials a resultant matrix (Sylvester or B´ezout) is computed and reduced to Cauchy-like form. The GKO algorithm of Gohberg, Kailath and Olshevsky [8] for the PLU factorization is applied to the Cauchy-like matrix obtained in this way. The algorithm relies on the pivoting strategy and on a suitable technique used to control the growth of the generators. This algorithm is rank-revealing since in exact arithmetic it provides a matrix U with the last k rows equal to zero, where k is the nullity of the matrix. In our case, where the computation is performed in ﬂoating point arithmetic with precision μ and where √ > μ, the algorithm is halted if the last computed pivot a is such that |a| ≤ m + n. This provides an estimate of the value k and a candidate g (x) to an -divisor of u(x) and v(x). In the second stage, the tentative -divisor g (x) is reﬁned by means of Newton’s iteration and a test is applied to check that g (x) is an -common divisor. In this part, the value of k can be adaptively modiﬁed in the case where g (x) has not the maximum degree, or if g (x) is not an -divisor of u(x) and v(x). It is important to point out that the Jacobian system, which has to be solved at each Newton’s iteration step, is still a Toeplitz-like linear system which can be reduced once again to Cauchy-like form and solved by means of the pivoted GKO algorithm. In the reﬁnement stage we have complemented Newton’s iteration with a line search step in order to guarantee the monotonic behavior of the residual. The algorithm has been implemented in Matlab, tested with a wide set of polynomials and compared with the currently available software, in particular the Matlab and Maple packages UVGCD by Zeng [23], STLN by Kaltofen et al. [14] and QRGCD by Corless et al. [4]. We did not compare our algorithm to the ones of [16], [24] since the software of the latter algorithms is not available. We have considered the test polynomials of [23] and some new additional tests which are representative of diﬃcult situations. In all the problems tested so far our algorithm has shown a high reliability and eﬀectiveness, moreover, its O(n2 ) complexity makes it much faster than the currently available algorithms already for moderately large values of the degree. Our Matlab code is available upon request. The paper is organized as follows. In Section 2 we recall the main tools used in the paper, among which, the properties of Sylvester and B´ezout matrices, their interplay with gcd, the reduction to Cauchy-like matrices and a modiﬁed version of the GKO algorithm. In Section 3 we present the algorithms for estimating the degree and the coeﬃcients of the -gcd together with the reﬁnement stage based on Newton’s iteration. Section 4 reports the results of the numerical experiments together with the comparison of our Matlab implementation with the currently available software.

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2. Resultant matrices and -gcd We recall the deﬁnitions of B´ezout and Sylvester matrices and their interplay with gcd. 2.1. Sylvester and B´ezout matrices The Sylvester matrix of u(x) and v(x) is the (m + n) × (m + n) matrix ⎞ ⎛ un un−1 . . . u0 0 ⎟ ⎜ .. .. ⎟ ⎜ . . ⎟ ⎜ ⎜ 0 un un−1 . . . u0 ⎟ ⎟. S(u, v) = ⎜ ⎜ vm vm−1 . . . v0 0 ⎟ ⎟ ⎜ ⎟ ⎜ .. .. ⎠ ⎝ . . 0

vm

vm−1

...

(1)

v0

where the coeﬃcients of u(x) appear in the ﬁrst m rows. Assume that n ≥ m and observe that the rational function u(x)v(y) − u(y)v(x) b(x, y) = x−y n i−1 j−1 is actually a polynomial i,j=1 x y bi,j in the variables x, y. The coeﬃcient matrix B(u, v) = (bi,j ) is called the B´ezout matrix of u(x) and v(x). The following property is well known: Lemma 2.1. The nullities of S(u, v) and of B(u, v) coincide with deg(g). The next two results show how the gcd of u(x) and v(x) and the corresponding cofactors are related to Sylvester and B´ ezout submatrices. Recall that, for an μ integer ν ≥ 2 and a polynomial a(x) = j=0 aj xj , the ν-th convolution matrix associated with a(x) is the Toeplitz matrix having [a0 , . . . , aμ , 0 . . . 0]T as its ﬁrst ν−1

column and [a0 , 0, . . . 0] as its ﬁrst row. ν−1

Lemma 2.2. Let u(x) = g(x)p(x), v(x) = g(x)q(x), then the vector [q0 , . . . , qm−k , −p0 , . . . , −pn−k ]T belongs to the null space of the matrix Sk = [Cu Cv ], where Cu is the (m − k + 1)-st convolution matrix associated with u(x) and Cv is the (n − k + 1)-st convolution matrix associated with v(x). Theorem 2.3. [6] Assume that B(u, v) has rank n − k and denote by c1 , . . . , cn its columns. Then ck+1 , . . . , cn are linearly independent. Moreover writing each ci (1 ≤ i ≤ k) as a linear combination of ck+1 , . . . , cn n ck−i = hk+1 c + hjk−i cj , i = 0, . . . , k − 1, k+1 k−i j=k+2

one ﬁnds that D(x) = d0 x + d1 x + · · · + dk−1 x + dk is a gcd for u(x) and v(x), where d1 , . . . , dk are given by dj = d0 hk+1 k−j+1 , with d0 ∈ R or C. k

k−1

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Moreover, we have: k ˆ(x) and Remark 2.4. Let g(x) = i=0 gi xi be the gcd of u(x) and v(x), and let u vˆ(x) be such that u(x) = u ˆ(x)g(x), v(x) = vˆ(x)g(x). Then we have B(u, v) = GB(ˆ u, vˆ)GT , where G is the (n − k)th convolution matrix associated with g(x). 2.2. Cauchy-like matrices An n × n matrix C is called Cauchy-like of rank r if it has the form ' ui vH (n−1 j , C= fi − a ¯j i,j=0

(2)

with ui and vj row vectors of length r ≤ n, and fi and aj complex scalars such that fi − a ¯j = 0 for all i, j. The matrix G whose rows are given by the ui ’s and the matrix B whose columns are given by the vi ’s are called the generators of C. Equivalently, C is Cauchy-like of rank r if the matrix ∇C C = F C − CAH ,

(3)

where F = diag(f0 , . . . , fn−1 ) and A = diag(a0 , . . . , an−1 ), has rank r. The operator ∇C deﬁned in (3) is a displacement operator associated with the Cauchy-like structure, and C is said to have displacement rank equal to r. The algorithm that we now present is due to Gohberg, Kailath and Olshevsky [8], and is therefore known as GKO algorithm; it computes the Gaussian elimination with partial pivoting (GEPP) of a Cauchy-like matrix and can be extended to other classes of displacement structured matrices. The algorithm relies on the following Fact 2.5. Performing Gaussian elimination on an arbitrary matrix is equivalent to applying recursive Schur complementation; Schur complementation preserves the displacement structure; permutations of rows and columns preserve the Cauchy-like structure. It is therefore possible to directly apply Gaussian elimination with partial pivoting to the generators rather than to the whole matrix C, resulting in increased computational speed and less storage requirements. So, a step of the fast GEPP algorithm for a Cauchy-like matrix C = C1 can be summarized as follows (we assume that generators (G1 , B1 ) of the matrix are given):

d1 (i) Use (2) to recover the ﬁrst column of C1 from the generators. l1 (ii) Determine the position (say, (k, 1)) of the entry of maximum magnitude in the ﬁrst column. (iii) Let P1 be the permutation matrix that interchanges the ﬁrst and kth rows. Interchange the ﬁrst and kth diagonal entries of F1 ; interchange the ﬁrst and kth rows of G1 .

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(iv) Recover from the generators the ﬁrst row d˜1 u1 of P1 C1 . Now one has % & 1 the ﬁrst column of L and the ﬁrst row d˜1 u1 of U in the LU 1 ˜ l d˜ 1 1

factorization of P1 C1 . (v) Compute generators (G2 , B2 ) of the Schur complement C2 of P1 · C1 as follows: % &

' ( 1 0 0 B2 = B1 − b1 1 d˜11 u1 , g1 , = G1 − (4) 1 ˜ l G2 d˜ 1 1

where g1 is the ﬁrst row of G1 and b1 is the ﬁrst column of B1 . Proceeding recursively, one obtains the factorization C1 = P · L · U , where P is the product of the permutation matrices used in the process. Now, let ⎞ ⎛ 0 ... ... 0 φ ⎜ 1 0 ... ... 0 ⎟ ⎟ ⎜ ⎜ .. ⎟ .. ⎟ . 0 1 . (5) Zφ = ⎜ ⎟ ⎜ ⎟ ⎜ . . . . .. . . .. ⎠ ⎝ .. 0 ... 0 1 0 and deﬁne the matrix operator ∇T T = Z1 T − T Z−1 .

(6)

An n × n matrix T having low displacement rank with respect to the operator ∇T (i.e., such that ∇T = GB, with G ∈ Cn×r and B ∈ Cr×n ) is called Toeplitz-like. Sylvester and B´ezout matrices are Toeplitz-like, with displacement rank 2. Toeplitz-like matrices can be transformed into Cauchy-like as follows [10]. Here and hereafter ˆı denotes the imaginary unit such that ˆı2 = −1. Theorem 2.6. Let T be an n × n Toeplitz-like matrix. Then C = F T D0−1 F H is a Cauchy-like matrix, i.e., ˆ B, ˆ ∇D1 ,D−1 (C) = D1 C − CD−1 = G

(7)

where 1 2πˆı F = √ [e n (k−1)(j−1) ]k,j n is the normalized n × n Discrete Fourier Transform matrix D1 = diag(1, e

ˆı

2π n

,...,e

ˆı(n−1)

2π n

),

(2n−1)πˆ ı πˆ ı 3πˆı D−1 = diag(e n , e n , . . . , e n ),

(n−1)πˆ ı πˆ ı D0 = diag(1, e n , . . . , e n ),

and ˆ = F G, G

ˆ H = F D0 B H . B

(8)

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Therefore the GKO algorithm can be also applied to Toeplitz-like matrices, provided that reduction to Cauchy-like form is applied beforehand. In particular, the generators (G, B) of the matrix S(u, v) with respect to the Toeplitz-like structure can be chosen as follows. Let N = n + m; then G is the N × 2 matrix having all zero entries except the entries (1, 1) and (m + 1, 2) which are equal to 1; the matrix B is 2 × N , its ﬁrst and second rows are [−un−1 , . . . , −u1 , vm − u0 , vm−1 , . . . , v1 , v0 + un ], [−vm−1 , . . . , −v1 , un − v0 , un−1 , . . . , u1 , u0 + vm ], respectively. Generators for B(u, v) can be similarly recovered from the representation of the B´ezout matrix as sum of products of Toeplitz/Hankel triangular matrices. Generators for the associated Cauchy-like matrix are computed from (G, B) by using (8).

2.3. Modiﬁed GKO algorithm Gaussian elimination with partial pivoting (GEPP) is usually regarded as a fairly reliable method for solving linear systems. Its fast version, though, raises more stability issues. Sweet and Brent [22] have done an error analysis of the GKO algorithm applied to a Cauchy-like matrix C. They point out that the error propagation depends not only on the magnitude of the triangular factors in the LU factorization of C (as is expected for ordinary Gaussian elimination), but also on the magnitude of the generators. In some cases, the generators can suﬀer large internal growth, even if the triangular factors do not grow too large, and therefore cause a corresponding growth in the backward and forward error. Experimental evidence shows that this is the case for Cauchy-like matrices derived from Sylvester and B´ezout matrices. However, it is possible to modify the GKO algorithm so as to prevent generator growth, as suggested for example in [21] and [9]. In particular, the latter paper proposes to orthogonalize the ﬁrst generator before each elimination step; this guarantees that the ﬁrst generator is well conditioned and allows a good choice of a pivot. In order to orthogonalize G, we need to: – QR-factorize G, obtaining G = GR, where G is an n × r column orthogonal matrix and R is upper triangular; ˜ = G and B ˜ = RB. – deﬁne new generators G This method performs partial pivoting on the column of C corresponding to the column of B with maximum norm. This technique is not equivalent to complete pivoting, but nevertheless allows a good choice of pivots and eﬀectively reduces element growth in the generators, as well as in the triangular factors.

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3. Fast -gcd computation 3.1. Estimating degree and coeﬃcients of the -gcd We ﬁrst examine the following problem: ﬁnd a fast method to determine whether two given polynomials u(x) and v(x) have an -divisor of given degree k. Throughout we assume that the input polynomials have unitary Euclidean norm. The coeﬃcients of the cofactors p(x) and q(x) can be obtained by applying Lemma 2.2. Once the cofactors are known, a tentative gcd can be computed as g(x) = u(x)/p(x) or g(x) = v(x)/q(x). Exact or nearly exact polynomial division (i.e., with a remainder of small norm) can be performed in a fast and stable way via evaluation/interpolation techniques (see [3]), which exploit the properties of the discrete Fourier transform. Alternatively, Theorem 2.3 can be employed to determine the coeﬃcients of a gcd; the cofactors, if required, are computed as p(x) = u(x)/g(x) and q(x) = v(x)/g(x). The matrix in Lemma 2.2 is formed by two Toeplitz blocks and has displacement rank 2 with respect to the straightforward generalization of the operator ∇T deﬁned in (6) to the case of rectangular matrices. We seek to employ the modiﬁed GKO algorithm to solve the system that arises when applying Lemma 2.2, or the linear system that yields the coeﬃcients of a gcd as suggested by Theorem 2.3. In order to ensure that the matrices F and A deﬁning the displacement operator ∇C associated with the reduced matrix have well-separated spectra, a modiﬁed version of Theorem 2.6 is needed. Observe that a Toeplitz-like matrix T also has low displacement rank with respect to the operator ∇Z1 ,Zθ (T ) = Z1 T − T · Zθ , for any θ ∈ C, |θ| = 1. Then we have: Theorem 3.1. Let T ∈ Cn×m be a Toeplitz-like matrix, satisfying ∇Z1 ,Zθ (T ) = Z1 T − T Zθ = GB, where G ∈ Cn×α , B ∈ Cα×m and Z1 , Zθ are as in (5). Let N = lcm (n, m). Then C = Fn T Dθ Fm is a Cauchy-like matrix, i.e., ˆ B, ˆ ∇D1 ,Dθ (C) = D1 C − CDθ = G

(9)

where Fn and Fm are the normalized Discrete Fourier Transform matrices of order n and m respectively, Dθ = θ · D1 , πˆ ı 2πˆ ı D = diag(1, e N m , e N m , . . . ) 2πˆ ı 2πˆ ı D1 = diag(1, e n , . . . , e n (n−1) )

ˆ = Fn G, and G

ˆ H = Fm DB H . B

πˆ ı The optimal choice for θ is then θ = e N .

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The gcd and cofactors obtained from Lemma 2.2 or Theorem 2.3 can be subsequently reﬁned as described in the next section. After the reﬁning step, it is easy to check whether an -divisor has actually been computed. We are left with the problem of choosing a tentative gcd degree k . A possibility is to employ a bisection technique, which requires to test the existence of an approximate divisor log2 n times and therefore preserves the overall quadratic cost of the method. Alternatively, a heuristic method of choosing a tentative value for k can be designed by observing that, as a consequence of the properties of resultant matrices presented in Section 2.1, the choice of a suitable k is mainly a matter of approximate rank determination, and the fast LU factorization of the Sylvester or B´ezout matrix might provide reasonably useful values for k . Observe that the incomplete fast LU factorization computes a Cauchy-like perturbation matrix ΔC such that C − ΔC has rank n − k. If a is the last pivot computed in the incomplete factorization, then as a consequence of Lemma 2.2 in [9], |a| ≤ ΔC2 . Now, let u (x) and v (x) be polynomials of minimum norm and same degrees as u(x) and v(x), such that u+u and v +v have an exact gcd of degree k. Assume u 2 ≤ and v 2 ≤ . Let C be the Cauchy-like matrix obtained via Theorem 2.6 from the Sylvester matrix S = S(u , v ). Then C + C has rank n − k, too. If we assume that ΔC2 is very close to the minimum norm of a Cauchy-like perturbation that decreases the rank of C to n − k, then we have √ (10) |a| ≤ ΔC2 ≤ C 2 = S 2 ≤ n + m, where the last inequality follows from the structure of the Sylvester matrix. There√ fore, if |a| > / n + m, then u(x) and v(x) cannot have an -divisor of degree k. This gives an upper bound on the -gcd degree based on the absolute values of the pivots found while applying the fast Gaussian elimination to C. The same idea can be applied to the B´ezout matrix. This is clearly a heuristic criterion since it assumes that some uncheckable condition on ||ΔC||2 is satisﬁed. However, this criterion seems to work quite well in practice. When it is applied, the gcd algorithm should check whether it actually provides an upper bound on the gcd degree. We use this criterion for the determination of a tentative gcd degree in our implementation of the algorithm. In fact, experimental evidence shows that this criterion is usually more eﬃcient in practice than the bisection strategy, though in principle it does not guarantee that the quadratic cost of the overall algorithm is preserved. 3.2. Reﬁnement Since the computed value of k is the result of a tentative guess, it might happen in principle that the output provided by the algorithm of Section 3.1 is not an -divisor, is an -divisor of lower degree, or is a poor approximation of the sought divisor. In order to get rid of this uncertainty, it is suitable to reﬁne this output by means of an ad hoc iterative technique followed by a test on the correctness

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of the -degree. For this purpose we apply Newton’s iteration to the least squares problem deﬁned by ⎤ ⎡

g Cp g − u (11) , z = ⎣ p ⎦, F (z) = Cq g − v q where the Euclidean norm of the function F (z) is to be minimized. Here, in boldface we denote the coeﬃcient vectors of the associated polynomials. The matrices Cp and Cq are convolution matrices of suitable size associated with the polynomials p(x) and q(x) respectively. The Jacobian matrix J associated with the problem (11) has the form

Cp C g 0 , (12) J= Cq 0 Cg where each block is a convolution matrix associated with a polynomial; Cp is of size (n+1)×(k+1), Cq is (m+1)×(k+1), Cg in the ﬁrst block row is (n+1)×(n−k+1) and Cg in the second block row is (m + 1) × (m − k + 1). This Jacobian matrix, however, is always rank deﬁcient, because of the lack of a normalization for the gcd. Remark 3.2. Under the hypotheses stated above, the Jacobian matrix (12) computed at any point z = [gT − pT − qT ]T is singular. Moreover, the nullity of J is 1 if and only if p(x), q(x) and g(x) have no common factors. In particular, if z is a solution of F (z) = 0 and g(x) has maximum degree, i.e., it is a gcd, then J has nullity one and any vector in the null space of J is a multiple of w = [gT − pT − qT ]T , where p(x) and q(x) are cofactors. In order to achieve better stability and convergence properties, we force the Jacobian to have full rank by adding a row, given by wT . Nevertheless, it can be proved, by relying on the results of [20], that the quadratic convergence of Newton’s method in the case of zero residual also holds, in this case, with a rank deﬁcient Jacobian. This property is useful when the initial guess for k is too small, since in this case the rank deﬁciency ' ( of the Jacobian is unavoidable. ˜ The new Jacobian J = wJT is associated with the least squares problem ' ( F (z) , where K is a constant. The choice that minimizes F˜ (z) = g2 −p 2 −q2 −K of wT as an additional row helps to ensure that the solution of each Newton’s step ˜ j )† F˜ (zj ) zj+1 = zj − J(z (13) ˜ j )† is the Moore-Penrose pseudoinverse of is nearly orthogonal to ker J. Here J(z ˜ j ). For ease of notation, the new Jacobian will be denoted simply the matrix J(z as J in the following. The matrix J has a Toeplitz-like structure, with displacement rank 5. We propose to exploit this property by approximating the solution of each linear least squares problem Jηj = F˜ (zj ), ηj = zj − zj+1

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via fast LU factorization still preserving the quadratic convergence of the modiﬁed Newton’s iteration obtained in this way. We proceed as follows: – Compute the factorization J = LU , where J ∈ CN ×M , L ∈ CN ×N and U ∈ CN ×M . For the sake of simplicity, we are overlooking here the presence of permutation matrices due to the pivoting procedure; we can assume that either J or the vectors ηj and xj = F˜ (zj ) have already undergone appropriate permutations. Consider the following block subdivision of the matrices L e U , where the left upper block has size M × M :

L1 0 U1 L= , U= . L2 I 0 % &

(1) xj L−1 0 1 −1 . Analogously, let xj = and observe that L = (2) −L2 L−1 I xj 1 – Let yj = L−1 1 xj . If U1 is nonsingular, then compute wj as solution of U1 wj = yj . Else, consider the block subdivision & & % %

(1) (1) wj yj U11 U12 , yj = , , wj = U1 = (2) (2) 0 0 wj yj (1)

(2)

such that U11 is nonsingular; set all the entries of wj (1) wj

(1) U11 wj

equal to zero, and

(1) yj

compute as solution of = – If J is rank deﬁcient, ﬁnd a basis for K = ker J. – Subtract from wj its projection on K, thus obtaining a vector χj . This is the vector that will be used as approximation of a solution of the linear least squares system in the iterative reﬁnement process. Let R be the subspace of CN spanned by the columns of J. We have CN = R ⊕ R⊥ .

(14)

Let xj = αj + βj be the decomposition of xj with respect to (14), i.e., we have αj ∈ R and βj ∈ R⊥ . The Moore-Penrose pseudoinverse of J acts on xj as follows: J † αj is the preimage of αj with respect to J and it is orthogonal to K = ker J, whereas J † βj is equal to zero. The LU-based procedure, on the other hand, acts exactly like J † on αj , whereas the component βj is not necessarily sent to 0. Therefore, χj is the sum of ηj and of the preimage of βj with respect to the LU decomposition. In a general linear least squares problem, there is no reason for βj 2 to be signiﬁcantly smaller than xj 2 . In our case, though, the Taylor expansion of F (z) yields: 0 = F (z∗ ) = F (zj ) − J(zj )j + O(j 22 ), (15)

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where j = zj − z∗ and z∗ is such that F (z∗ ) = 0. It follows from (15) that xj = J(zj )j + O(j 22 ). Since J(zj )j ∈ R, we conclude that βj 2 = O(j 22 ). Therefore, Newton’s method applied to the iterative reﬁnement of the polynomial gcd preserves its quadratic convergence rate, even though the linear least squares problems (13) are treated using via the LU factorization of the Jacobian. The iterative process ends when at least one of the following criteria is satisﬁed: 1. the residual (that is, the Euclidean norm of the function F (z)) becomes smaller than a ﬁxed threshold, 2. the number of iteration reaches a ﬁxed maximum, 3. the residual given by the last iteration is greater than the residual given by the previous iteration. The purpose of the third criterion is to avoid spending computational eﬀort on tentative gcds that are not in fact suitable candidates. However, its use with Newton’s method may pose some diﬃculties, because it is generally diﬃcult to predict the global behaviour of this method; in particular, it might happen that the residual does not decrease monotonically. The usual way to overcome this obstacle is to use instead a relaxed version of Newton that includes a line search. More precisely, instead of the iteration (13) one computes ˜ j )† F˜ (zj ), zj+1 = zj − αj J(z

(16)

where αj is chosen – using a one-dimensional minimization method – so as to approximately minimize the norm of F˜ (zj ). The drawback of this technique is that it slows down convergence: the quadratic convergence that was one of the main interesting points of Newton’s method is lost if one consistently performs iterations of the type (16). For this reason we employ here a hybrid method: At each step, the algorithm evaluates the descent ˜ j )† F˜ (zj ) and checks if a pure Newton step (that is, (16) with αj = 1) direction J(z decreases the residual. If this is the case, then the pure Newton step is actually performed; otherwise, αj and subsequently zj+1 are computed by calling a line search routine. In this way, most of the optimization work is still performed by pure Newton iterations, so that the overall method remains computationally cheap; the line search, called only when necessary, is helpful in some diﬃcult cases and ensures that the method has a sound theoretical basis. 3.3. The overall algorithm Algorithm Fastgcd Input: the coeﬃcients of polynomials u(x) and v(x) and a tolerance . Output: an -gcd g(x); a backward error (residual of the gcd system); possibly perturbed polynomials u ˆ(x) and vˆ(x) and cofactors p(x) and q(x). Computation: – Compute the Sylvester matrix S associated with u(x) and v(x); – Use Lemma 2.6 to turn S into a Cauchy-like matrix C;

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– Perform fast Gaussian elimination√with almost complete pivoting on C; stop when a pivot a such that |a| < / n + m is found; let k0 be the order of the ˜ that has a as upper left entry; not-yet-factored submatrix U – Choose k = k0 as tentative gcd degree; – Is there an -divisor of degree k? The answer is found as follows: – ﬁnd tentative cofactors by applying the modiﬁed GKO algorithm to the system given by Lemma 2.2, – compute a tentative gcd by performing polynomial division via evaluation/interpolation, – perform iterative reﬁnement and check whether the backward error is smaller than ; – If yes, check for k+1; if there is also an -divisor of degree k+1, keep checking for increasing values of the degree until a maximum is reached (i.e., a degree is found for which there is no -divisor); – If not, keep checking for decreasing values of the degree, until an -divisor (and gcd) is found. Observe that a slightly diﬀerent version of the above algorithm is still valid by replacing the Sylvester matrix with the B´ezout matrix. With this replacement the size of the problem is roughly reduced by a factor of 2 with clear computational advantage. It should also be pointed out that the algorithm generally outputs an approximate gcd with complex coeﬃcients, even if u(x) and v(x) are real polynomials. This usually allows for a higher gcd degree or a smaller backward error.

4. Numerical experiments The algorithm Fastgcd has been implemented in Matlab and tested on many polynomials, with satisfactory results. Some of these results are shown in this section and compared to the performance of other implemented methods that are found in the literature, namely UVGCD by Zeng [23], STLN by Kaltofen et al. [14] and QRGCD by Corless et al. [4]. Matlab experiments with Fastgcd and UVGCD are performed using version 7.5.0 running under Windows; we use here the P-code for UVGCD contained in the Apalab toolbox. It must be pointed out that comparison with the STLN method is not straightforward, since this methods follows an optimization approach, i.e., it takes two (or more) polynomials and the desired gcd degree k as input, and seeks a perturbation of minimum norm such that the perturbed polynomials have an exact gcd of degree k. Moreover, the algorithms UVGCD and STLN do not normalize the input polynomials, whereas QRGCD and Fastgcd do; therefore all test polynomials are normalized (with unitary Euclidean norm) beforehand. In the following tests, we generally display the residual (denoted as “res”) associated with the gcd system (recall that the residual is deﬁned here as the Euclidean norm of the function F (z) and it may slightly diﬀer from the residual as

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deﬁned by other authors). In some examples, where a nearly exact gcd is sought, we report the coeﬃcient-wise error on the computed gcd (denoted as “cwe”), since the “correct” gcd is known. 4.1. Badly conditioned polynomials The test polynomials in this section are taken from [23]. The polynomials in the ﬁrst example are speciﬁcally chosen so that the gcd problem is badly conditioned. Example 4.1. Let n be an even positive integer and k = n/2; deﬁne pn = un vn and qn = un wn , where un = wn =

k

[(x − r1 αj )2 + r12 βj2 ],

j=1 n

vn =

k

[(x − r2 αj )2 + r22 βj2 ],

j=1

[(x − r1 αj )2 + r12 βj2 ],

αj = cos

j=k+1

jπ , n

βj = sin

jπ , n

for r1 = 0.5 and r2 = 1.5. The roots of pn and qn lie on the circles of radius r1 and r2 . The following table shows the coeﬃcient-wise errors given by the examined gcd methods as n increases. n

Fastgcd

UVGCD

QRGCD

10 12 14 16 18

6.44 × 10−13 5.23 × 10−12 1.79 × 10−11 5.27 × 10−10 6.11 × 10−9

3.24 × 10−13 1.40 × 10−12 2.27 × 10−11 4.41 × 10−11 3.63 × 10−10

1.57 × 10−12 3.28 × 10−4 (∗) (∗) (∗)

(∗) Here QRGCD fails to ﬁnd a gcd of correct degree.

In this case, there are no substantial diﬀerences between the (good) results provided by Fastgcd and by UVGCD, while QRGCD outputs failure for very illconditioned cases. It should be pointed out, however, that the results given by UVGCD vary between trials, which makes comparisons more diﬃcult. In the following test, the gcd degree is very sensitive to the choice of the tolerance . Example 4.2. Let p(x) =

10 (x − xj ), 1

j

q(x) =

10

(x − xj + 10−j ),

1

with xj = (−1) (j/2). The roots of p and q have decreasing distances 0.1, 0.01, 0.001, etc.

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169

The table shows, for several values of the tolerance, the corresponding gcd degree and residual found by Fastgcd and UVGCD. Fastgcd gives better results, since it generally ﬁnds gcds of higher degree. The algorithm QRGCD, on the contrary, outputs failure for all values of smaller than 10−2 . Fastgcd UVGCD 10−2 10−3 10−4 10−6 10−7

deg

res

deg

res

9 8 7 6 5

0.0045 2.63 × 10−4 9.73 × 10−6 2.78 × 10−7 8.59 × 10−9

9 8 (∗) 1 1

0.0040 1.72 × 10−4 3.34 × 10−16 3.34 × 10−16

(∗) Here UVGCD outputs the same result as above due to a diﬀerent deﬁnition of residual.

It is interesting to observe that for ≤ 10−5 , UVGCD computes a common -divisor which does not have the maximum degree, while Fastgcd always provides an -divisor of higher degree. We have also studied this example using the STLN method, though the employed approach is entirely diﬀerent. The following table shows the residuals computed by STLN for several values of the degree. deg gcd res deg gcd res 9 8 7

5.65 × 10−3 2.44 × 10−4 1.00 × 10−5

6 5 4

2.58 × 10−7 6.34 × 10−9 1.20 × 10−10

4.2. High gcd degree In this example, also taken from [23], the gcd has a large degree. 3 Example 4.3. Let pn = un v and qn = un w, where v(x) = j=0 xj and w(x) = 4 j j=0 (−x) are ﬁxed polynomials and un is a polynomial of degree n whose coefﬁcients are random integer numbers in the range [−5, 5]. The following table shows the residuals and the coeﬃcient-wise errors on the computed gcd for large values of n. Here, Fastgcd and UVGCD perform similarly while QRGCD provides a worse coeﬃcient-wise error. n 50 100 200 500 1000

Fastgcd

UVGCD

QRGCD

res

cwe

res

cwe

cwe

2.97 × 10−16 2.91 × 10−16 5.08 × 10−16 4.04 × 10−16 3.98 × 10−16

5.04 × 10−16 1.41 × 10−15 7.29 × 10−15 3.12 × 10−15 3.28 × 10−15

2.43 × 10−16 1.83 × 10−16 1.72 × 10−16 2.10 × 10−15 2.26 × 10−16

8.32 × 10−16 7.77 × 10−16 9.99 × 10−16 1.35 × 10−14 1.67 × 10−15

1.72 × 10−12 4.80 × 10−8 2.39 × 10−11

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4.3. Unbalanced coeﬃcients This is another example taken from [23]. Example 4.4. Let p = uv and q = uw, where v(x) and w(x) are as in Example 4.3 and 15 u(x) = cj 10ej xj , j=0

where cj and ej are random integers in [−5, 5] and [0, 6] respectively. In this example u(x) is the gcd of p(x) and q(x) and the magnitude of its coeﬃcients varies between 0 and 5×106 . If an approximate gcd algorithm is applied and the coeﬃcient-wise relative error θ is calculated, then N = log10 θ is roughly the minimum number of correct digits for the coeﬃcients of u(x) given by the chosen method. 100 repetitions of this test are performed. The average number of correct digits found in an experiment of this type is 10.63 for Fastgcd and 10.83 for UVGCD. Therefore the two algorithms give comparable results. Residuals are always about 10−16 . QRGCD, on the contrary, achieves an average of 7.46 correct digits. 4.4. Multiple roots Example 4.5. Let u(x) = (x3 + 3x − 1)(x − 1)k for a positive integer k, and let v(x) = u (x). The gcd of u(x) and v(x) is g(x) = (x − 1)k−1 . The coeﬃcient-wise errors computed by Fastgcd, UVGCD and QRGCD for several values of k and for = 10−6 are shown in the following table. Unless otherwise speciﬁed, the computed gcd degrees are understood to be correct. k Fastgcd UVGCD QRGCD 15 5.18 × 10−13 25 9.31 × 10−11 35 1.53 × 10−8 45 6.61 × 10−6

4.27 × 10−13 1.99 × 10−11 4.44 × 10−9 4.04 × 10−8

7.04 × 10−7 (∗) (∗) (∗)

(∗) Here QRGCD does not detect a gcd of correct degree.

The algorithm UVGCD has been speciﬁcally designed for polynomials with multiple roots and is therefore very eﬃcient. Fastgcd also provides good results, with backward errors (residuals) always of the order of the machine epsilon, whereas QRGCD fails to ﬁnd a gcd of correct degree as soon as the root multiplicity is larger than Donnerstag, Oktober 8, 2009 at 3:44 pm15. 4.5. Small leading coeﬃcient A gcd with a small leading coeﬃcient may represent in many cases a source of instability. Example 4.6. For a given (small) parameter α ∈ R, let g(x) = αx3 + 2x2 − x + 5, p(x) = x4 + 7x2 − x + 1 and q(x) = x3 − x2 + 4x − 2 and set u(x) = g(x)p(x), v(x) = g(x)q(x).

A Fast Algorithm for Approximate Polynomial GCD

171

6 5

[ ZŦ

4

log (time)

3 0 Ŧ Ŧ Ŧ 3.5

4

4.5

5

5.5

6

6.5

7

log (N)

Figure 1. Running time of the algorithm Fastgcd We applied Fastgcd and QRGCD to this example, with α ranging between 10−5 and 10−15 . It turns out that, for α < 10−5, QRGCD fails to recognize the correct gcd degree and outputs a gcd of degree 2. Fastgcd, on the contrary, always recognizes the correct gcd degree, with a residual of the order of the machine epsilon. 4.6. Running time We have checked the growth rate of the running time of the algorithm Fastgcd on pairs of polynomials whose GCD and cofactors are deﬁned like the polynomials un (x) introduced in Section 4.2. Polynomials of degree N = 2n ranging between 50 and 1000 have been used. Figure 1 shows the running time (in seconds) versus the degree in log-log scale, with a linear ﬁt and its equation. Roughly speaking, the running time grows as O(N α ), where α is the coeﬃcient of the linear term in the equation, i.e., 2.06 in our case. We next show a comparison between the running times of Fastgcd and UVGCD. In order to avoid randomly chosen coeﬃcients, we deﬁne a family of test polynomials as follows. Let k be a positive integer and let n1 = 25k, n2 = 15k and n3 = 10k. For each value of k deﬁne the cofactors pk (x) = (xn1 −1)(xn2 −2)(xn3 −3) and qk (x) = (xn1 +1)(xn2 +5)(xn3 +ˆı). The test polynomials are uk (x) = g(x)pk (x) and vk (x) = g(x)qk (x), where the gcd g(x) = x4 +10x3 +x−1 is a ﬁxed polynomial. Figure 2 shows the computing times required by Fastgcd and UVGCD on uk (x) and vk (x) for k = 1, . . . , 8. The plot clearly shows that the time growth for Fastgcd is much slower than for UVGCD.

172

D.A. Bini and P. Boito 90 Fastgcd UVGCD

80 70

time

60 50 40 30 20 10 0

1

2

3

4

5

6

7

8

k

Figure 2. Comparison between the running times of Fastgcd and UVGCD.

References [1] B. Beckermann, G. Labahn, When are two numerical polynomials relatively prime?, J. Symbolic Comput. 26, 677–689 (1998). [2] B. Beckermann, G. Labahn, A fast and numerically stable Euclidean-like algorithm for detecting relatively prime numerical polynomials, J. Symb. Comp. 26, 691–714 (1998). [3] D.A. Bini, V.Y. Pan, Polynomial and Matrix Computations, vol. I, Birkh¨ auser, 1994. [4] R.M. Corless, S.M. Watt, L. Zhi, QR Factoring to Compute the GCD of Univariate Approximate Polynomials, IEEE Trans. Signal Processing 52, 3394–3402 (2004). [5] R.M. Corless, P.M. Gianni, B.M. Trager, S.M. Watt, The Singular Value Decomposition for Approximate Polynomial Systems, Proc. International Symposium on Symbolic and Algebraic Computation, July 10–12 1995, Montreal, Canada, ACM Press 1995, pp. 195–207. [6] G.M. Diaz-Toca, L. Gonzalez-Vega, Computing greatest common divisors and squarefree decompositions through matrix methods: The parametric and approximate cases, Linear Algebra Appl. 412, 222–246 (2006). [7] I.Z. Emiris, A. Galligo, H. Lombardi, Certiﬁed approximate univariate GCDs, J. Pure Appl. Algebra 117/118, 229–251 (1997). [8] I. Gohberg, T. Kailath, V. Olshevsky, Fast Gaussian elimination with partial pivoting for matrices with displacement structure, Math. Comp. 64, 1557–1576 (1995).

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[9] M. Gu, Stable and Eﬃcient Algorithms for Structured Systems of Linear Equations, SIAM J. Matrix Anal. Appl. 19, 279–306 (1998). [10] G. Heinig, Inversion of generalized Cauchy matrices and other classes of structured matrices, Linear Algebra in Signal Processing, IMA volumes in Mathematics and its Applications 69, 95–114 (1994). [11] G. Heinig, P. Jankowsky, K. Rost, Fast inversion of Toeplitz-plus-Hankel matrices, Numer. Math. 52, 665–682 (1988). [12] V. Hribernig, H.J. Stetter, Detection and validation of clusters of polynomial zeros, J. Symb. Comp. 24, 667–681 (1997). [13] C.-P. Jeannerod, G. Labahn, SNAP User’s Guide, UW Technical Report no. CS2002-22 (2002). [14] E. Kaltofen, Z. Yang, L. Zhi, Approximate Greatest Common Divisors of Several Polynomials with Linearly Constrained Coeﬃcients and Singular Polynomials, Proc. International Symposium on Symbolic and Algebraic Computations, 2006. [15] N.K. Karmarkar, Y.N. Lakshman, On Approximate GCDs of Univariate Polynomials, J. Symbolic Comp. 26, 653–666 (1998). [16] B. Li, Z. Yang, L. Zhi, Fast Low Rank Approximation of a Sylvester Matrix by Structure Total Least Norm, Journal of Japan Society for Symbolic and Algebraic Computation 11, 165–174 (2005). [17] M.-T. Noda, T. Sasaki, Approximate GCD and its application to ill-conditioned algebraic equations, J. Comput. Appl. Math. 38, 335–351 (1991). [18] V.Y. Pan, Numerical computation of a polynomial GCD and extensions, Information and Computation 167, 71–85 (2001). [19] A. Sch¨ onhage, Quasi-GCD Computations, J. Complexity, 1, 118–137 (1985). [20] L.B. Rall, Convergence of the Newton Process to Multiple Solutions, Num. Math. 9, 23–37 (1966). [21] M. Stewart, Stable Pivoting for the Fast Factorization of Cauchy-Like Matrices, preprint (1997). [22] D.R. Sweet, R.P. Brent, Error analysis of a fast partial pivoting method for structured matrices, in Adv. Signal Proc. Algorithms, Proc. of SPIE, T. Luk, ed., 266–280 (1995). [23] Z. Zeng, The approximate GCD of inexact polynomials Part I: a univariate algorithm, to appear [24] L. Zhi, Displacement Structure in computing the Approximate GCD of Univariate Polynomials, Mathematics, W. Sit and Z. Li eds., World Scientiﬁc (Lecture Notes Series on Computing), 288–298 (2003). Dario A. Bini and Paola Boito Dipartimento di Matematica Universit` a di Pisa Largo Bruno Pontecorvo 5 I-56127 Pisa, Italy e-mail: [email protected] [email protected]

Operator Theory: Advances and Applications, Vol. 199, 175–189 c 2010 Birkh¨ auser Verlag Basel/Switzerland

On Inertia of Some Structured Hermitian Matrices Vladimir Bolotnikov Dedicated to the memory of Georg Heinig

Abstract. Two classes of structured Hermitian matrices are considered with the additional property that certain principal submatrices are all singular. Such matrices can be considered as the Pick matrices of certain (interior and boundary) norm constrained interpolation problems for functions meromorphic on the unit disk which the iterative Schur algorithm does not apply to. We characterize these matrices in terms of the parameters determining their structure and present formulas for their inertia. Mathematics Subject Classiﬁcation (2000). 15A57, 11C20. Keywords. Structured Hermitian matrices, inertia.

1. Introduction In this note we discuss two related classes of structured matrices whose structure is determined by Stein equations and the additional property that the principal minors of certain type are all zeros. Let Jn (z) denote the n × n Jordan block with z ∈ C on the main diagonal and let En stand for the vector of the length n with the ﬁrst coordinate equals one and other coordinates equal zero: ⎡

z

⎢ ⎢ 1 Jn (z) = ⎢ ⎢ ⎣ 0

0 z .. .

⎤ ... 0 . ⎥ .. . .. ⎥ ⎥, ⎥ .. . 0 ⎦ 1 z

⎡ ⎢ ⎢ En = ⎢ ⎣

1 0 .. . 0

⎤ ⎥ ⎥ ⎥. ⎦

(1.1)

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V. Bolotnikov

Given a tuple z = (z1 , . . . , zk ) of k distinct points in the open unit disk D and a tuple n = (n1 , . . . , nk ) of natural numbers, we let |n| := n1 + · · · + nk and deﬁne ⎤ ⎤ ⎡ ⎡ Jn1 (z1 ) En1 ⎥ ⎢ ⎢ . ⎥ .. Tn (z) = ⎣ (1.2) ⎦ and En = ⎣ .. ⎦ . . Jnk (zk )

Enk

Deﬁnition 1.1. We say that a matrix P ∈ C|n|×|n| belongs to the class Dn if it satisﬁes the Stein identity P − Tn (z)P Tn (z)∗ = En En∗ − Cn Cn∗ for some vector

⎤ C1,n1 ⎥ ⎢ .. Cn = ⎣ ⎦, . Ck,nk ⎡

⎡ where

ci,0 .. .

⎢ Ci,ni = ⎣

(1.3) ⎤ ⎥ ⎦,

(1.4)

ci,ni −1 ∗

and if its compression to any Tn (z) -invariant subspace of C|n| is singular. Since all the eigenvalues of Tn (z) fall inside the unit disk, the Stein equation (1.3) has a unique solution P which is Hermitian. Solving (1.3) gives the following explicit formulas for the entries of P : k

P = [Pij ]i,j=1 ,

Pij ∈ Cni ×nj ,

(1.5)

where

min{ ,r}

[Pij ] ,r

=

s=0

−

zir−s z¯j −s ( + r − s)! ( − s)!s!(r − s)! (1 − zi z¯j ) +r−s+1

r min{α,β}

α=0 β=0

s=0

(1.6)

β−s α−s ∗ (α + β − s)! zi z¯j ci, −α cj,r−β . (α − s)!s!(β − s)! (1 − zi z¯j )α+β−s+1

In (1.6) and in what follows we will use the symbol c∗ for the complex conjugate of c ∈ C. Also we will denote by π(P ), ν(P ) and δ(P ) respectively the numbers of positive, negative and zero eigenvalues, counted with multiplicities, of a Hermitian matrix P . For two tuples n and m in Zk+ , we will say that m = (m1 , . . . , mk ) (n1 , . . . , nk ) = n if mi ≤ ni

for i = 1, . . . , k.

(1.7)

For a matrix P = Pn decomposed in blocks as in (1.5) and a tuple m n as in k (1.7), deﬁne the principal submatrix Pm = [(Pm )ij ]i,j=1 of P whose block entries (Pm )ij ’s are equal to the leading mi × mj submatrices of the corresponding blocks in P :

Imj k . (1.8) Pm = [(Pm )ij ]i,j=1 where (Pm )ij = Imi 0 Pij 0 It is easily seen that any compression of P to a Tn (z)∗ -invariant subspace of C|n| is of the form Pm for some m n. Thus the class Dn can be characterized as follows.

On Inertia of Some Structured Hermitian Matrices

177

Deﬁnition 1.1 . Dn consists of all matrices P of the form (1.5), (1.6) (for some ci,j ∈ C) and such that for every m n, the submatrix Pm of P is singular. Classes Dn admit the following functional-model interpretation. Let H 2 be the Hardy space of the unit disk, let θ(z) be the ﬁnite Blaschke product

ni k z − zi θ(z) = 1 − z z¯i i=1 associated with the tuples z and n, and let Kθ := H 2 θH 2 be the model space. The functions

1 dj 1 − θ(z)θ(zi )∗ ei,j (z) = (i = 1, . . . , k; j = 0, . . . , ni − 1), (1.9) j! d¯ zi 1 − z z¯i form a basis for Kθ and therefore, dim Kθ = |n|. The space Kθ is invariant with (0) respect to the backward shift operator R : f → f (z)−f and the matrix of R z ∗ with respect to the basis (1.9) is equal to Tn (z) deﬁned in (1.2). Let Tf be the Toeplitz operator with symbol f ∈ H ∞ and let Pθ,f be the compression of the operator I − Tf Tf∗ to the model space Kθ . Using the reproducing property of the 1 − θ(z)θ(ζ)∗ kernel for the space Kθ , it is not hard to show that the matrix of 1 − z ζ¯ the operator Pθ,f with respect to the basis (1.9) equals ⎡⎡ ⎤r=0,...,nj −1 ⎤k 1 1

+r 1 − f (z)f (ζ) 1 ⎢ 1 ∂ ⎥ ⎦ Pnf (z) = ⎣⎣ . (1.10) 1 ⎦ !r! ∂z ∂ ζ¯r 1 − z ζ¯ 1 z=zi ζ=zj

=0,...,ni −1

i,j=1

If in addition, f (j) (zi ) = j! ci,j

(i = 1, . . . , k; j = 0, . . . , ni − 1),

(1.11)

then diﬀerentiation in (1.10) shows that Pnf (z) is equal to the matrix P deﬁned in (1.5), (1.6). Furthermore, P ∈ Dn means that the compression of Pθ,f to any backward shift invariant subspace of Kθ is singular. Now we will explain why the class Dn (or equivalently, the class of operators on Kθ of the form PKθ (I −Tf Tf∗ )|Kθ with all backward shift invariant compressions singular) are of some interest. Note that the matrix Pnf (z) can be deﬁned for every function f analytic at z1 , . . . , zk (not necessarily analytic on all of D). If we will think of (1.11) as of interpolation conditions for an unknown function f (say, rational and with maxz∈T |f (z)| ≤ 1), then Pnf (z) = P for every solution f of the problem (1.11) with interpolation data {zi , ci,j }. The matrix P is then called the Pick matrix of the problem (1.11). In the nondegenerate case (where P is invertible), the solution set of the problem (1.11) can be parametrized in terms of a linear fractional transformation

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V. Bolotnikov

(see, e.g., [4]). If P is singular, one may start with a subproblem of (1.11) of the form f (j) (zi ) = j! ci,j (i = 1, . . . , k; j = 0, . . . , mi − 1), (1.12) where mi ≤ ni for i = 1, . . . , k. It is readily seen that the Pick matrix of this subproblem is the principal submatrix Pm of P deﬁned as in (1.8), and if this matrix is invertible, one can apply the Schur algorithm to reduce the original problem (1.11) to a problem with |m| fewer interpolation conditions. We now see that the class Dn consists of Pick matrices corresponding to interpolation problems of the form (1.11) which the Schur algorithm (even the ﬁrst step of this algorithm) does not apply to. On the other hand, even if we start with a degenerate interpolation problem (1.11) containing nondegenerate subproblems, then after a number of steps we still come up with a reduced problem whose Pick matrix belongs to the class Dm for some m n. In other words, the Schur algorithm reduces any degenerate problem (1.11) to a problem with the Pick matrix from the class Dm . In Section 2 we characterize the matrices of the class Dn in terms of the parameters {ci,j } and establish a simple formula for the inertia of a P ∈ Dn (the results of this sort go back to [8]; see also [1], [2], [3] for further developments). The formula for inertia (see Theorem 2.2) is of certain interest for interpolation problems by rational functions unimodular on the unit circle T (i.e., ratios of ﬁnite Blaschke products: f = b1 /b2 ), since the inertia of the Pick matrix controls the minimally possible degrees of b1 and b2 . In Section 3 we consider the class Bn (a “boundary” analog of the class Dn ), the class of matrices with leading (with respect to a designated block decomposition) minors equal zero and with the structure determined by the Stein identity (1.3) where the (distinct) points t1 , . . . , tk fall on the unit circle T rather than inside the unit disk. In this case, for the Stein equation (1.3) to have a solution, the vector Cn must satisfy certain additional a’priori conditions. If these conditions are satisﬁed, the equation has inﬁnitely many Hermitian solutions, and all of them are of the same special structure. We will see how this structure simpliﬁes under the singularity assumption about leading submatrices. The matrices of the class Bn serve as Pick matrices of boundary L∞ -norm constrained interpolation problems which do not contain nondegenerate subproblems. Applications of the results presented here to boundary interpolation problems will be demonstrated elsewhere.

2. The interior case Since the unique solution P of the Stein equation (1.3) is determined by z1 , . . . , zk and the entries ci,j of the vector Cn , the characterization of matrices from the class Dn can be given in terms of this data. It turns out that (distinct) points z1 , . . . , zk do not play any role in this characterization. Proposition 2.1. A matrix P of the form (1.5), (1.6) belongs to Dn if and only if c1,0 = c2,0 = · · · = ck,0 = γ,

|γ| = 1

(2.1)

On Inertia of Some Structured Hermitian Matrices and i = 1, . . . , k and 1 ≤ j ≤

179

'n ( i

, (2.2) 2 where [x] stands for the greatest integer less than or equal to x. Proof. Let for short i = n2i and let us assume that P ∈ Dn . Then in particular, the leading submatrices of diagonal blocks Pii are singular. Since Pii satisﬁes the Stein identity ci,j = 0

for every

∗ Pii − Jni (zi )Pii Jni (zi )∗ = Eni En∗ i − Ci,ni Ci,n i

(2.3)

(which is just the equality between the ith diagonal blocks in (1.7)), Theorem 5.1 in [1] applies and says that the i × i leading submatrix P ii of Pii is the zero matrix. Now we compare the i × i leading submatrices in (2.3) to conclude that ∗ 0 = E i E ∗i − Ci, i Ci,

where E i and Ci, i are deﬁned via formulas (1.1) and (1.4). i Equating the top rows in the latter equality gives 0 = 1 − |ci,0 |2 −ci,0 c∗i,1 −ci,0 c∗i,2 . . . ci,0 c∗i, , which implies (2.4) |ci,0 | = 1 for i = 1, . . . , k and ci,1 = ci,2 = · · · = ci, i = 0 which is the same as (2.2). Now we take the tuple m = (m1 , . . . , mk ) with mi = mj = 1 and m = 0 for = i, j. Since m n, the submatrix ⎤ ⎡ ⎤ ⎡ 1 − ci,0 c∗j,0 1 − |ci,0 |2 1 − ci,0 c∗j,0 0 ⎢ 1 − |zi |2 ⎢ 1 − zi z¯j ⎥ 1 − zi z¯j ⎥ ⎥ ⎥=⎢ Pm = ⎢ ∗ 2 ⎦ ⎣ 1 − cj,0 ci,0 1 − |cj,0 | ⎦ ⎣ 1 − cj,0 c∗i,0 0 2 1 − zj z¯i 1 − zj z¯i 1 − |zj | of P is singular, which implies ci,0 = cj,0

for

i, j = 1, . . . , k.

(2.5)

Equalities (2.1) follow from (2.4) and (2.5). For the suﬃciency part, conditions (2.1) and (2.2) guarantee observe that n that all the entries in the ( n2i + 1) × ( 2j + 1) leading submatrix of the block Pij of P are zeros (this is readily seen from the explicit formula (1.6) for the entries of Pij ): % & 0([ ni ]+1)×([ nj ]+1) ∗ 2 2 Pij = (i, j = 1, . . . , k). (2.6) ∗ ∗ Since Pij ∈ Cni ×nj , it follows that the dimensions of the zero block in (2.6) are greater than halves of the corresponding dimensions of Pij . Therefore, for every m n, the matrix Pm contains a zero principal submatrix whose dimensions are greater than |m| 2 . Therefore, Pm is singular which completes the proof of the theorem. The next theorem describes the inertia of a matrix P ∈ Dn . It shows in particular, that a matrix P ∈ Dn has equally many positive and negative eigenvalues.

180

V. Bolotnikov

Note that in the single-block case (i.e., if k = 1) this theorem is a particular case of Theorem 5.1 in [1]. Theorem 2.2. Let z = (z1 , . . . , zk ) ∈ Dk and let n = (n1 , . . . , nk ) ∈ Nk and d = (d1 , . . . , dk ) ∈ Zk+ be two tuples such that ni ≤ di ≤ ni for i = 1, . . . , k. (2.7) 2 Let P be deﬁned as in (1.5), (1.6) and let us assume that the numbers ci,j satisfy (2.1) and ci,di = 0

and

ci,j = 0

i = 1, . . . , k and 1 ≤ j ≤ di − 1

for every

(2.8)

(the ﬁrst inequality in (2.8) is relevant only if di < ni ). Then π(P ) = ν(P ) = |n| − |d|

and

δ(P ) = 2|d| − |n|.

(2.9)

Proof. Note that under assumptions (2.8), the matrix P belongs to Dn if and only if n2i < di ≤ ni for i = 1, . . . , k (by Proposition 2.1). However, the conclusion (2.9) of the theorem holds true even if di = n2i for some i’s. In particular, it follows that if di = n2i for every i = 1, . . . , k, then the matrix P is invertible and has equally many positive and negative eigenvalues. For the proof, we ﬁrst observe that conditions (2.1) and (2.8) guarantee that Pd , the principal submatrix of P deﬁned via (1.8), is equal to the zero matrix Pd = 0.

(2.10)

The latter can be seen directly from the explicit formula (1.6) for the entries of P . Since P ∈ C|n|×|n| and Pd ∈ C|d|×|d| , it follows from (2.10), that rank(P ) ≤ 2(|n| − |d|) and therefore, that δ(P ) ≥ |n| − 2(|n| − |d|) = 2|d| − |n| ≥ 0,

(2.11)

where the last inequality follows from the assumption (2.7). j=0,...,ni −1 j=0,...,2di −1 to {ci,j }i=1,...,k ; the extendLet us extend the sequence {ci,j }i=1,...,k ing terms ci,ni +r (r = 0, . . . , 2di − ni − 1) are arbitrary nonzero complex numbers (in fact we need only ci,ni to be nonzero in case where ni = di ). Let H be the solution of the Stein equation ∗ ∗ ∗ H − T2d HT2d = E2d E2d − C2d C2d

(2.12)

where 2d := (2d1 , . . . , 2dk ) and where the matrices T2d := T2d (z), E2d and C2d are deﬁned via formulas (1.2) and (1.4). It is clear that the principal submatrix Hn of H is equal to the original matrix P . We will show that π(H) = ν(H) = |d|.

(2.13)

Assuming that (2.13) is already proved, we complete the proof of the theorem as follows. Since P is an |n| × |n| principal submatrix of the 2|d| × 2|d| Hermitian matrix H, it follows by the Cauchy’s interlacing theorem that π(P ) ≥ π(H) − (2|d| − |n|)

and ν(P ) ≥ ν(H) − (2|d| − |n|)

On Inertia of Some Structured Hermitian Matrices

181

which together with (2.13) imply π(P ) ≥ |n| − |d| ≥ 0

and ν(P ) ≥ |n| − |d| ≥ 0.

(2.14)

Since π(P ) + ν(P ) + δ(P ) = |n|, equalities (2.9) follow from inequalities (2.11) and (2.14). It remains to verify (2.13). To this end, let ⎤ ⎡ ⎤ ⎡ ci,di 0 ... 0 ci,di ⎢ .. ⎥ .. ⎢ ci,di +1 ⎥ ⎢ ci,di +1 ci,di . . ⎥ ⎥ ⎢ ⎥ , (2.15) ⎢ and F Fi,di := ⎢ := ⎥ .. i,di ⎥ ⎢ . .. .. ⎦ ⎣ .. . ⎣ . . 0 ⎦ ci,2di −1 ci,2di −1 . . . ci,di +1 ci,di ⎡

⎡

⎤ F1,d1 ⎢ ⎥ Fd = ⎣ ... ⎦

and let

⎢ and Fd = ⎣

Fk,dk

F1,d1

0 ..

.

0

⎤ ⎥ ⎦.

(2.16)

Fk,dk

Ci,di and since Ci,di = γEdi (by Since C2d = Col1≤i≤k Ci,2di , since Ci,2di = Fi,di assumptions (2.1) and (2.8)), we have eventually

γEdi . (2.17) C2d = Col1≤i≤k Fi,di

U1 where U1 and Let U be the 2|d| × |2d| permutation matrix deﬁned by U = U2 U2 are diagonal block matrices with the ith diagonal blocks U1,i = Idi 0di and U2,i = 0di Idi . Then

T U T2d U = d ∗ ∗

0 , Td

U E2d

Ed = , 0

U C2d

γEd = , Fd

(2.18)

where the two ﬁrst equalities follow from deﬁnitions (1.2) and the third is a consequence of (2.17). We also have (by (2.10))

B∗ P 0 B∗ (2.19) U HU ∗ = d = (B, D ∈ C|d|×|d| ). B D B D Multiplying both parts of (2.12) by U on the left and by U ∗ on the right and making use of (2.18) and (2.19), we get

Td 0 0 B 0 B Td∗ ∗ 0 −γEd Fd∗ − = . ∗ Td B ∗ D 0 Td∗ −γ ∗ Fd Ed∗ −Fd Fd∗ B∗ D Comparison the 21-blocks in the latter matrix equality gives B − Td BTd∗ = −γ ∗ Fd Ed∗ . |d|×|d|

Let Q ∈ C

(2.20)

be the unique solution of the Stein equation Q − Td QTd∗ = Ed Ed∗ .

(2.21)

182

V. Bolotnikov

Since the pair (Ed , Td ) is observable, the matrix Q is positive deﬁnite. Multiplying both parts of (2.21) on the left by the matrix Fd given in (2.16) and taking into account that Td Fd = Fd Td and Fd Ed = Fd , we arrive at Fd Q − Td Fd QTd∗ = Ed Ed∗ . Comparing the latter equality with (2.20) we conclude that the matrix −γFd Q and B solve the same Stein equation. Since this equation has a unique solution, it follows that B = −γFd Q. Since γ = 0 and since the matrices Q and Fd are not singular (recall that the diagonal entries of the triangular matrix Fd are all nonzero either by assumption (2.8) or by construction), it follows that B is not singular. Then the matrix U HU ∗ in (2.19) is invertible and therefore, it has equally many positive and negative eigenvalues (this is a direct consequence of the Cauchy’s interlacing theorem). Thus, π(U HU ∗ ) = ν(U HU ∗ ) = |d|, and (2.13) follows, which completes the proof of the theorem. Corollary 2.3. Let P belong to Dn . Then π(P ) =

k

π(Pii ),

i=1

ν(P ) =

k i=1

ν(Pii ),

δ(P ) =

k

δ(Pii ).

(2.22)

i=1

Proof. Applying (2.9) separately to each diagonal block Pii we get π(Pii ) = ν(Pii ) = ni − di

and δ(Pii ) = 2di − ni

for i = 1, . . . , k which together with (2.9) (applied to the whole P ) imply the statement. Corollary 2.4. If P ≥ 0 (P ≤ 0) belongs to Dn , then P = 0. Proof. The statement follows from (2.9), since in this case ν(P ) = 0 (π(P ) = 0). Corollary 2.5. Let n = (n1 , . . . , nk ) ∈ Nk with ni ≤ 2 for i = 1, . . . , k. If P ∈ Dn , then P = 0. Proof. By Theorem 2.2, equalities (2.1) hold and moreover, ci,1 = 0 whenever ni = 2. Then the right-hand side matrix in the Stein equation (1.3) is the zero matrix and the result follows since the homogeneous Stein equation P − Tn (z)P Tn (z)∗ = 0 has only trivial solution.

3. The boundary case In this section we consider the Stein equation (1.3) where Tn (z), En and Cn are still given by formulas (1.2), (1.4), but the (distinct) points t1 , . . . , tk fall on the unit circle T rather than inside the unit disk: P − Tn (t)P Tn (t)∗ = En En∗ − Cn Cn∗ ,

(3.1)

where t = (t1 , . . . , tk ) ∈ T . In this “boundary” situation, the Stein equation (3.1) either has no solutions or it has inﬁnitely many of them. In the latter case, the structure of every solution P is determined by |n| additional parameters which k

On Inertia of Some Structured Hermitian Matrices

183

will be now explained (we refer to [5, Section 10], [6, Section 3] and [7, Section 2] for proofs and some more detail). Given t ∈ T and n ∈ N, let n ⎤ ⎡ (−1)n−1 n−1 t −t2 t3 · · · 0 t ⎢ 0 −t3 2t4 · · · (−1)n−1 n−1 tn+1 ⎥ 1 ⎥ ⎢ ⎢ .. n+2 ⎥ 5 n−1 n−1 ⎥, ⎢ · · · (−1) t t (3.2) Ψn (t) = ⎢ . 2 ⎥ ⎥ ⎢ . . . . . . ⎦ ⎣ . . . 2n−1 t 0 ··· ··· 0 (−1)n−1 n−1 n−1 be the upper triangular matrix with the entries Ψj = (−1) j t +j+1 for 0 ≤ j ≤ ≤ n − 1. Let Ci,ni be the lower triangular Toeplitz matrix ⎡ ⎤ 0 ... 0 ci,0 ⎢ ci,1 ci,0 . . . 0 ⎥ ⎢ ⎥ i = 1, . . . , k, (3.3) Ci,ni = ⎢ .. ⎥ , .. . . . . ⎣ . . . ⎦ . ci,ni −1 . . . ci,1 ci,0 so that Ci,ni Eni = Ci,ni . A necessary and suﬃcient condition for the Stein equation (3.1) to have a solution is that ∗ C i,ni Ψni (ti )Ci,ni = Ψni (ti )

where gives

for i = 1, . . . , k,

(3.4)

stands for the transpose. Equating the left upper corner entries in (3.4) |ci,0 | = 1

Sequences so that

ni −1 {ci,j }j=0

for i = 1, . . . , k.

(3.5)

satisfying (3.7) can be extended (not uniquely) to

∗ C i,2ni Ψ2ni (ti )Ci,2ni = Ψ2ni (ti )

2ni −1 {ci,j }j=0

(i = 1, . . . , k)

(3.6)

and in general, (3.4) and (3.5) follow from (3.6) due to upper triangular structure of matrices in (3.6). The next theorem (in a slightly diﬀerent formulation) can be found in [6]. i −1 Theorem 3.1. Given t = (t1 , . . . , tk ) ∈ Tk and {ci,j }j=0,...,2n , let i=1,...,k

k

P = [Pij ]i,j=1

(3.7)

be the block matrix with the ni × nj blocks Pij given by

min{ ,r}

[Pij ] ,r

=

s=0

−

t¯ −s tr−s ( + r − s)! i j ( − s)!s!(r − s)! (1 − ti t¯j ) +r−s+1

r min{α,β} α=0 β=0

s=0

(3.8)

β−s α−s (α + β − s)! ti t¯j ci, −α c∗j,r−β (α − s)!s!(β − s)! (1 − ti t¯j )α+β−s+1

if

i = j

and Pii := Hi,ni Ψni (ti )C∗i,ni

where

n −1

i Hi,ni = [ci, +j+1 ] ,j=0

(i = 1, . . . , k). (3.9)

184

V. Bolotnikov

The following are equivalent 1. Conditions (3.6) hold. 2. P is Hermitian and satisﬁes the Stein identity (3.1). 3. P is Hermitian and |ci,0 | = 1 for i = 1, . . . , k. Moreover, if equation (3.1) has a solution (i.e., if conditions (3.4) are in force), then every Hermitian solution P is necessarily of the form (3.7)–(3.9) for some ci,j (i = 1, . . . , k; j = ni , . . . , 2ni − 1) satisfying extended conditions (3.6). Hermitian matrices of the form (3.7)–(3.9) can be generated by rational functions unimodular on T. Given such a function f and given a tuple t = (t1 , . . . , tk ) ∈ T, let us deﬁne the boundary Schwarz-Pick matrix by Pnf (t) = lim Pnf (z) z→t

(3.10)

where z ∈ Dk and Pnf (z) is deﬁned as in (1.10). The proof of the next theorem can be found in [7]. Theorem 3.2. Let t = (t1 , . . . , tk ) ∈ Tk and let f be a rational function unimodular on T. Then the numbers ci,j =

f (j) (ti ) j!

(i = 1, . . . , k; j = 0, . . . , 2ni − 1)

satisfy conditions (3.6). Furthermore, the limit in (3.10) exists and is equal to the matrix P deﬁned by formulas (3.7)–(3.9). Now we address the questions from Section 2 to the present boundary setting. Deﬁnition 3.3. We will say that P of the form (3.7)–(3.9) and with parameters {ci,j } satisfying conditions (3.6) belongs to the class Bn if for every m n, the principal submatrix Pm of P deﬁned as in (1.8) is singular. The next theorem is an analog of Proposition 2.1. In contrast to the interior case, the matrices of the class Bn are block diagonal. Proposition 3.4. Let t = (t1 , . . . , tk ) ∈ Tk and n = (n1 , . . . , nk ) ∈ Nk , let P be deﬁned as in (3.7)–(3.9) and let us assume that conditions (3.6) are satisﬁed. Then P ∈ Bn if and only if (2.1) holds and ci,j = 0

for every

i = 1, . . . , k and 1 ≤ j ≤ ni .

(3.11)

In this case, the block entries of P simplify to Pii = γHi,ni Ψni (ti )

and

Pij = 0 (i = j).

(3.12)

Proof. Since conditions (3.6) are satisﬁed, then in particular, we have (3.5) and therefore, the matrices Ci,j deﬁned via formula (3.3) are invertible for every i = 1, . . . , k and j = 0, . . . , ni − 1. Since ti ∈ T, the triangular matrices Ψj (ti ) are also invertible.

On Inertia of Some Structured Hermitian Matrices

185

Let P ∈ Bn . Then Pii ∈ Bni (i.e., all leading submatrices of Pii are singular) for every i = 1, . . . , k. It follows from the “Hankel-Ψ-Toeplitz” structure (3.9) of Pii that the j × j leading submatrix of Pii equals Hi,j Ψj (ti )C∗i,j and therefore, the membership of Pii in Bni implies that det Hi,j = 0

for every j = 0, . . . , ni − 1.

(3.13)

Now we recursively get (3.11) from (3.13). Indeed, letting j = 0 in (3.13) we get ci,1 = 0; assuming that ci,j = 0 for some j ∈ {1, . . . , ni } and ci,r = 0 for r = 1, . . . , j − 1 we get ⎡ ⎤ 0 ... 0 ci,j ⎢ .. ⎥ . . ⎢ . .. .. ci,j+1 ⎥ ⎢ ⎥ det Hi,j = det ⎢ .. ⎥ = 0, . . . . ⎣ 0 . . . ⎦ ci,j ci,j+1 . . . ci,2j−1 which contradicts (3.13) and proves (3.11). Equalities (2.1) are obtained in much the same way as in the proof of Theorem 2.1. We take the tuple m = (m1 , . . . , mk ) with mi = mj = 1 and m = 0 for = i, j. Since the leading entries in the blocks Pii and Pjj are equal to zeros, the submatrix Pm takes the form ⎤ ⎡ 1 − ci,0 c∗j,0 0 ⎢ 1 − ti t¯j ⎥ ⎥. Pm = ⎢ ⎦ ⎣ 1 − cj,0 c∗i,0 0 1 − tj t¯i Since P ∈ Bn , the matrix Pm must be singular which implies that ci,0 = cj,0 for i, j = 1, . . . , k. The latter implies (2.1) due to (3.5). On account of (2.1) and (3.11), Ci,ni = γIni and therefore, formula (3.9) for Pii collapses to the ﬁrst formula in (3.12). On the other hand, substituting (2.1) and (3.11) into (3.8) gives

min{ ,r}

[Pij ] ,r

=

s=0

min{ ,r}

=

s=0

r−s −s ( + r − s)! ti t¯j (1 − ci,0 c∗j,0 ) ( − s)!s!(r − s)! (1 − ti t¯j ) +r−s+1 r−s −s ( + r − s)! ti t¯j (1 − |γ|2 ) =0 ( − s)!s!(r − s)! (1 − ti t¯j ) +r−s+1

for every ≤ ni and r ≤ nj which proves the second equality in (3.12). Finally, let us assume that conditions (2.1) and (3.11) hold. Then representations (3.12) are in force. Furthermore, since ci,1 = · · · = ci,ni = 0 and since the matrix Ψni (ti ) is upper triangular, it follows from (3.12) that all the entries of Pii on and above the main “southwest-northeast” diagonal are zeroes. Therefore every leading submatrix of Pii is singular and thus, Pii ∈ Bni for every i = 1, . . . , k. Since all non-diagonal blocks in P are zero matrices, P ∈ Bn .

186

V. Bolotnikov

The next theorem is the “boundary” analog of Theorem 2.2. It establishes explicit formulas for inertia of a matrix P ∈ Bn of the form (3.7)–(3.9). Since every such P is necessarily block diagonal (by (3.12)), it suﬃces to consider the one-block case. We will use the previous notation with the subscript i dropped. Theorem 3.5. Let t ∈ T, let c0 , . . . , c2n−1 be the complex numbers such that ∗ C 2n Ψ2n (t)C2n = Ψ2n (t)

(3.14)

Pn := Hn Ψn (t)C∗n ,

(3.15)

and let where the matrices Hn and Cn are constructed via formulas (3.9) and (3.3). Let cd = 0

and

c1 = c2 = · · · = cd−1 = 0

for some d (n < d ≤ 2n − 1).

(3.16)

Then δ(Pn ) = d − n. Furthermore, 1. If d = 2k, then π(P ) = ν(P ) = n − k. 2. If d = 2k + 1, then td cd c∗0 ∈ R \ {0} and (a) ν(Pn ) = π(Pn ) + 1 = n − k if (−1)k sgn (td cd c∗0 ) < 0. (b) π(Pn ) = ν(Pn ) + 1 = n − k if (−1)k sgn (td cd c∗0 ) > 0. Proof. Condition (3.14) implies |c0 | = 1 and therefore, Cn = c0 In so that formula (3.15) reads (3.17) Pn = c∗0 Hn Ψn (t). Let Dj (A) (j = 1, . . . , 2n − 1) denote the jth “southwest-northeast” diagonal of a matrix A ∈ Cn×n . With a self-evident interpretation, 2 (aj,1 , aj−1,2 , . . . , a1,j ) , if 1 ≤ j ≤ n, Dj (A) = (an,j−n+1 , an−1,j−n+2 , . . . , aj−n+1,n ) , if n + 1 ≤ j ≤ 2n − 1. Since Ψn (t) is invertible for every t = 0 and n ∈ N, it follows from (3.17) that rank (Pn ) = rank (Hn ). Assumptions (3.16) and the Hankel structure of Hn imply that Dj (Hn ) = 0 (j = 0, . . . , d − 1) and Dd (Hn ) = (cd , cd , . . . , cd ) = 0. Since d > n, it now follows that rank (Hn ) = 2n − d and therefore, δ(Pn ) = δ(Hn ) = n − (2n − d) = d − n.

(3.18)

2d−1 Let us extend the sequence {ci }2n−1 i=0 to {ci }i=0 in such a way that ∗ C 2d Ψ2d (t)C2d = Ψ2d (t),

(3.19)

where C2d and Ψ2d are deﬁned via formulas (3.3) and (3.2). Such an extension is always possible (see [7, Section 2], where these extensions were called t-isometric). Now we can introduce the Hankel matrix Hd := [ci+j+1 ]d−1 i,j=0 and the matrix Pd := Hd Ψd (t)C∗d

(3.20)

On Inertia of Some Structured Hermitian Matrices which is Hermitian due to (3.19)

Hn ∗ Cn , Cd = Hd = ∗ ∗ 0

187

by Theorem 3.1. It is readily checked that

∗ ∗ Ψn (t) ∗ P , Ψd (t) = , Pd = n . ∗ 0 ∗ ∗ ∗

Since all the entries above the main diagonal Dd (Hd ) of Hd are zeros, it follows that det Hd = (−1)d−1 cdd . (3.21) ∗ Since the matrices Ψd (t) and Cd are upper triangular, their determinants are equal to products of diagonal entries, so that det Ψd (t) =

d−1

(−1)j t2j+1 = (−1)

d(d−1) 2

td

2

and

det C∗d = (c∗0 )d ,

j=0

which together with (3.21) and (3.20) gives det Pd = (−1)

(d+2)(d−1) 2

(td c∗0 cd )d .

(3.22)

By triangular structure of Hd , Ψd (t) and C∗d , we conclude from (3.20) that Dj (Pd ) = 0 for j = 1, . . . , d − 1. Note also that Pd is Hermitian and invertible: det Hd = 0, by (3.21). Therefore, if d = 2k, then Pd is of the form

0 B Pd = , B ∈ Ck×k , det B = 0. B∗ D Therefore (3.23) π(Pd ) = ν(Pd ) = k. Since Pn ∈ Cn×n is a principal submatrix of Pd ∈ Cd×d , it follows by the Cauchy’s interlacing theorem that π(Pn ) ≥ π(Pd ) − (d − n) and ν(Pn ) ≥ ν(Pd ) − (d − n).

(3.24)

Since d = 2k < 2n, the latter inequalities and equalities (3.23) imply π(Pn ) ≥ k − (2k − n) = n − k > 0 and ν(Pn ) ≥ n − k > 0.

(3.25)

Since π(Pn ) + ν(Pn ) + δ(Pn ) = n, it follows from (3.18) and (3.25) that in fact, π(Pn ) = n − k and ν(Pn ) = n − k. Finally, let us assume that d = 2k + 1 and let pkk be the “central” entry of Pd . Since Pd is Hermitian, pkk ∈ R and Pd takes the form ⎡ ⎤ 0 0 B (3.26) Pd = ⎣ 0 pkk X ⎦ , where B ∈ Ck×k and X ∈ C1×k . B∗ X ∗ D Since Pd is invertible, we have det B = 0 and pkk = 0. Now we will show that sgn (pkk ) = sgn (det Pd ) = (−1)k sgn (tk cd c∗0 ).

(3.27)

Indeed, since the matrix B in (3.26) is upper triangular with respect to its main “southwest-northeast” diagonal Dk (B), we have det Pd = pkk · | det B|2 and the ﬁrst equality in (3.27) follows. To verify the second, let us equate the entries on

188

V. Bolotnikov

the intersection of the ﬁrst row and of the (d + 1)-st column in matrix equality (3.19); by deﬁnitions (3.2) and (3.3) of matrices C2d and Ψ2d (t) we have c0 c1 . . . c2d Ψ2d (t) c∗0 c∗1 . . . c∗d 0 . . . 0 = (−1)d−1td+2 , which in view of (3.2), (3.16) collapses to tc0 c∗d + (−1)d t2d+1 cd c∗0 + (−1)d−1 td+2 |c0 |2 = (−1)d−1td+2 . Since d is odd and since |c0 | = 1, we get tc0 c∗d −t2d+1 cd c∗0 = 0. Multiplying the latter equality by t¯d+1 and taking into account that |t| = 1, we get t¯d c0 c∗d − td cd c∗0 = 0. Therefore, td cd c∗0 ∈ R and since d is odd, sgn (tk cd c∗0 ) = sgn (tk cd c∗0 )d .

(3.28)

Substituting d = 2k + 1 into (3.22) gives det Pd = (−1)k(2k+3) (td c∗0 cd )d = (−1)k (td c∗0 cd )d which together with (3.28) implies the second equality in (3.27). To complete the proof of the theorem, observe that the Schur complement S of pkk in Pd equals

0 B 0 B 0 −1 0 X S= − p = B∗ D X ∗ kk B ∗ D − X ∗ p−1 kk X and has equally many positive and negative eigenvalues by the preceding analysis. Since π(Pd ) = π(pkk ) + π(S) and ν(Pd ) = ν(pkk ) + ν(S), it follows that Pd has k positive eigenvalues, k negative eigenvalues and one more eigenvalue whose sign coincides with that of pkk . According to (3.28), π(Pd ) = ν(Pd ) − 1 = k

if

(−1)k sgn (td cd c∗0 ) < 0

(3.29)

ν(Pd ) = π(Pd ) − 1 = k

if (−1)k sgn (td cd c∗0 ) > 0.

(3.30)

and Now we again apply the interlace theorem to get inequalities (3.24) which then turn out to be equalities: π(Pn ) = π(Pd ) − (d − n) and ν(Pn ) = ν(Pd ) − (d − n), and then the second statement of the theorem follows from (3.29) and (3.30). Note that statement (1) also covers the trivial case d = 2n (which was not included in (3.16)). In this case Pn = 0 which agrees with statement (1) asserting that π(Pn ) = ν(Pn ) = n − n = 0. In conclusion we note that Corollary 2.5 does not hold for P ∈ Bn , whereas Corollary 2.3 does (which is obvious due to the block diagonal structure of P ∈ Bn ). As for Corollary 2.4, we have the following boundary analog which we formulate for the single-block case and which follows immediately from Theorem 3.5: If P ≥ 0 (P ≤ 0) belongs Bn , then rank(P ) ≤ 1.

On Inertia of Some Structured Hermitian Matrices

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References [1] D. Alpay and H. Dym, Structured invariant spaces of vector-valued rational functions, Hermitian matrices, and a generalization of the Iohvidov laws, Linear Algebra Appl. 137/138 (1990), 137–181. [2] D. Alpay and H. Dym, Structured invariant spaces of vector-valued functions, sesquilinear forms, and a generalization of the Iohvidov laws, Linear Algebra Appl. 137/138 (1990), 413–451 [3] D. Alpay and H. Dym, On a new class of reproducing kernel spaces and a new generalization of the Iohvidov laws, Linear Algebra Appl. 178 (1993), 109–183. [4] J.A. Ball, I. Gohberg, and L. Rodman, Interpolation of rational matrix functions, OT45, Birkh¨ auser Verlag, 1990. [5] V. Bolotnikov and H. Dym, On boundary interpolation for matrix Schur functions, Mem. Amer. Math. Soc. 181 (2006), no. 856. [6] V. Bolotnikov and A. Kheifets, The higher-order Carath´ eodory–Julia theorem and related boundary interpolation problems, Operator Theory: Advances and Applications OT 179 (2007), 63–102. [7] V. Bolotnikov and A. Kheifets, Carath´ eodory-Julia type conditions and symmetries of boundary asymptotics for analytic functions on the unit disk, Math. Nachr. 282 (2009), no. 11, 1513–1536. [8] I.S. Iohvidov, Hankel and Toeplitz matrices and forms. Algebraic theory. Birkh¨ auser, Boston, Mass., 1982. Vladimir Bolotnikov Department of Mathematics The College of William and Mary Williamsburg, VA 23187-8795, USA e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 199, 191–202 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Variable-coeﬃcient Toeplitz Matrices with Symbols beyond the Wiener Algebra Albrecht B¨ottcher and Sergei Grudsky In Memory of Georg Heinig

Abstract. Sequences of so-called variable-coeﬃcient Toeplitz matrices arise in many problems, including the discretization of ordinary diﬀerential equations with variable coeﬃcients. Such sequences are known to be bounded if the generating function satisﬁes a condition of the Wiener type, which is far away from the minimal requirement in the case of constant coeﬃcients. The purpose of this paper is to uncover some phenomena beyond the Wiener condition. We provide counterexamples on the one hand and prove easy-to-check suﬃcient conditions for boundedness on the other. Mathematics Subject Classiﬁcation (2000). Primary 47B35; Secondary 15A60, 65F35. Keywords. Toeplitz matrix, variable coeﬃcients, matrix norm.

1. Introduction Let a be a complex-valued continuous function on [0, 1] × [0, 1] × T, where T is the complex unit circle, a : [0, 1] × [0, 1] × T → C, (x, y, t) → a(x, y, t). For n ∈ Z, we 3 put a ˆn (x, y) = T a(x, y, t) t−n |dt|/(2π) and so have the Fourier series a(x, y, t) =

∞

a ˆn (x, y)tn ,

(1)

n=−∞

where equality holds at least in the L2 sense. Let AN (a) be the matrix

N j k AN (a) = a ˆj−k . , N N j,k=0

(2)

Occasionally we allow us to write a(x, y, t) and AN (a(x, y, t)) for a and AN (a). This work was partially supported by CONACYT project U46936-F, Mexico.

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We refer to AN (a) as a Toeplitz matrix with variable coeﬃcients. Clearly, if a does not depend on x and y, then AN (a) is a pure Toeplitz matrix. It is easily seen that every (N + 1) × (N + 1) matrix may be written as AN (a) with some a. Thus, the notion of a Toeplitz matrix with variable coeﬃcients is rather an asymptotic notion which makes sense for the entire sequences {AN (a)}∞ N =0 but not for a single (N + 1) × (N + 1) matrix. Variable-coeﬃcient Toeplitz matrices and their modiﬁcations and generalizations are currently emerging in many applications (see, for example, [1], [4], [5], [6], [7], [8], [11], [12]) and go under various names, such as generalized Toeplitz [9], locally Toeplitz [11] or generalized locally Toeplitz matrices [8], Berezin-Toeplitz matrices [1], twisted Toeplitz matrices [12], or generalized discrete convolutions [10], [13]. In our opinion, variable-coeﬃcient Toeplitz matrices [3] is the perhaps best name, at least when considering matrices of the form (2). The problem of primary interest is the understanding of the spectral and pseudospectral properties of AN (a) as N → ∞. Accordingly, the works cited above and also [3], [9], [14] deal with extensions of the Szeg¨ o and Avram-Parter theorems, that is, with asymptotic formulas for tr f (AN (a)), and with asymptotic inverses and pseudospectra of AN (a). This paper addresses the problem of the uniform boundedness of AN (a), that is, the question whether AN (a)∞ remains bounded as N → ∞. Here and in what follows, A∞ is the spectral norm of A. ∞ ˆn tn , we denote AN (a) by If a is independent of x and y, a(t) = n=−∞ a aj−k )N TN (a). Thus, TN (a) = (ˆ j,k=0 . It is well known that TN (a)∞ ≤ TN +1 (a)∞ for all N and that lim TN (a)∞ = M∞ (a) := sup |a(t)|.

N →∞

(3)

t∈T

If a is of the form a(x, y, t) = b(x, y)tn , then

N j+n j , AN (a) = TN (tn ) diag b N N j=0 (where j + n is taken modulo N + 1) and hence AN (b(x, y)tn )∞ ≤ M∞,∞ (b) := sup

sup |b(x, y)|.

x∈[0,1] y∈[0,1]

Consequently, for a given by (1) we have AN (a)∞ ≤

∞

M∞,∞ (ˆ an ).

(4)

n=−∞

Thus, if ∞ n=−∞

M∞,∞ (ˆ an ) < ∞,

(5)

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then {AN (a)∞ }∞ N =0 is a bounded sequence. Condition (5) is a condition of the Wiener type. In the case where a does not depend on x and y, it amounts to saying that TN (a) ∞ remains bounded as N → ∞ if a belongs to the Wiener algebra, that is, if ∞ an | < ∞. This is clearly far away from (3) and is therefore a n=−∞ |ˆ source of motivation for looking whether the uniform boundedness of AN (a)∞ can be guaranteed under weaker assumptions. The ﬁrst question one might ask is whether the sole continuity of the generating function a on [0, 1]2 × T := [0, 1] × [0, 1] × T ensures the uniform boundedness of AN (a)∞ . We show that, surprisingly, the answer to this question is no. Theorem 1.1. There exist a ∈ C([0, 1]2 × T) such that sup AN (a)∞ = ∞. On the other hand, we will prove that AN (a)∞ remains bounded if a(x, y, t) has certain smoothness in x and y. (Notice that (5) is a requirement on the smoothness in t.) Our results will imply the following. Theorem 1.2. If a ∈ C 4,0 ([0, 1]2 × T), which means that the function a(x, y, t) has continuous partial derivatives with respect to x and y up to the order 4, then sup AN (a)∞ < ∞. For a(x, y, t) independent of x and y, this is equivalent to the statement that TN (a) is uniformly bounded if a ∈ C(T). In fact, we can sharpen Theorem 1.2 as follows. Theorem 1.3. If a ∈ L∞ (T, C 4 ([0, 1]2 )), that is, if a is an L∞ function on T with values in the Banach space of all functions on [0, 1]2 that have continuous partial derivatives up to the order 4, then sup AN (a)∞ < ∞. In the case of constant coeﬃcients, this theorem is best possible: it says that sup TN (a)∞ < ∞ if a ∈ L∞ (T). The next question is whether the exponent 4 is close to a kind of a minimum. Let 0 < α ≤ 1. For a continuous function a(x, y, t) on [0, 1]2 × T we deﬁne Mα,∞,∞ (a) = sup sup

sup

t∈T y∈[0,1] x1 ,x2

M∞,α,∞ (a) = sup sup sup

t∈T x∈[0,1] y1 ,y2

|a(x2 , y, t) − a(x1 , y, t)| , |x2 − x1 |α |a(x, y2 , t) − a(x, y1 , t)| , |y2 − y1 |α

and

|Δ2 a(x1 , x2 , y1 , y2 , t)| α α t∈T x1 ,x2 y1 ,y2 |x2 − x1 | |y2 − y1 | where Δ2 a(x1 , x2 , y1 , y2 , t) is the second diﬀerence Mα,α,∞ (a) = sup sup sup

Δ2 a(x1 , x2 , y1 , y2 , t) = a(x2 , y2 , t) − a(x2 , y1 , t) − (a(x1 , y2 , t) − a(x1 , y1 , t)). Here supz1 ,z2 means the supremum over all z1 , z2 ∈ [0, 1] such that z1 = z2 . We say that the function a(x, y, t) belongs to Hα,α,∞ if the three numbers Mα,α,∞ (a),

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Mα,∞,∞ (a), M∞,α,∞ (a) are ﬁnite and we denote by H1+α,1+α,∞ the set of all functions a(x, y, t) that have continuous partial derivatives up to the order 2 in x and y and for which the three numbers M1+α,∞,∞ (a) := Mα,∞,∞ (∂x a), M∞,1+α,∞ (a) := M∞,α,∞ (∂y a), M1+α,1+α,∞ (a) := Mα,α,∞ (∂x ∂y a) are ﬁnite. Notice that C 4,0 ([0, 1]2 × T) ⊂ H2,2,∞ . We here prove the following. Theorem 1.4. If β < 1/2, there exist functions a in the space Hβ,β,∞ such that sup AN (a)∞ = ∞. If β > 1, then sup AN (a)∞ < ∞ for every function a in Hβ,β,∞ . The theorem leaves a gap. We can actually remove this gap and prove that sup AN (a)∞ < ∞ whenever a ∈ Hβ,β,∞ and β > 1/2, which is guaranteed if a ∈ C 2,0 ([0, 1]2 × T). However, the proof of this result is very sophisticated. We see the purpose of this paper in revealing the delicacy of the problem of the uniform boundedness of AN (a)∞ and in providing results that might be suﬃcient for applications. Drawing down things to β > 1/2 is a matter of mathematical ambition and will be the topic of our subsequent paper [2].

2. H¨older continuity We already deﬁned the space Hβ,β,∞ and the quantities Mβ,β,∞ (a), Mβ,∞,∞ (a), M∞,β,∞ (a) for a continuous function a(x, y, t) on [0, 1]2 × T and for 0 < β ≤ 2. In addition, we put M∞,∞,∞ (a) = sup sup

sup |a(x, y, t)|.

t∈T x∈[0,1] y∈[0,1]

Note that if a(x, y, t) = xγ + y γ with 0 < γ < 1, then Mβ,∞,∞ (a) = M∞,β,∞ (a) = ∞,

Mβ,β,∞ (a) = 0

for γ < β < 1, which shows that the assumption Mβ,β,∞(a) < ∞ does not imply that Mβ,∞,∞ (a) and M∞,β,∞ (a) are ﬁnite. In the introduction we also mentioned that C 2,0 ([0, 1]2 × T) is contained in H1,1,0 and thus in Hβ,β,0 for 0 < β < 1. This follows from the representations 4 x2 a(x2 , y, t) − a(x1 , y, t) = ∂x a(ξ, y, t)dξ, x 4 y12 a(x, y2 , t) − a(x, y1 , t) = ∂y a(x, η, t)dη, y 4 x21 4 y2 ∂x ∂y a(ξ, η, t)dηdξ, Δ2 a(x1 , x2 , y1 , y2 , t) = x1

y1

which show that in fact the continuity of the partial derivatives ∂x a, ∂y a, ∂x ∂y a would suﬃce. Since C 2,0 ([0, 1]2 × T) ⊂ H1,1,0 , it follows that C 4,0 ([0, 1]2 × T) is a subset of H2,2,0 .

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In what follows we work with functions a(x, y, t) that are independent of one of the variables x and y and therefore need the following modiﬁcations. Let 0 < α ≤ 1. We say that a continuous functions a(x, t) on [0, 1] × T is in Hα,∞ if Mα,∞ (a) := sup sup t∈T x1 ,x2

|a(x2 , t) − a(x1 , t)| < ∞, |x2 − x1 |α

and we denote by M∞,∞ (a) the maximum of |a(x, t)| on [0, 1] × T. The function a(x, t) is said to be in H1+α,∞ if it is continuously diﬀerentiable in x and ∂x a is in Hα,∞ . In that case M1+α,∞ (a) is deﬁned as Mα,∞ (∂x a). Analogously, we say that a continuous functions a(y, t) on [0, 1] × T is in Hα,∞ if Mα,∞ (a) := sup sup t∈T y1 ,y2

|a(y2 , t) − a(y1 , t)| < ∞, |y2 − y1 |α

we denote by M∞,∞ (a) the maximum of |a(y, t)| on [0, 1] × T, and say that a(y, t) belongs to H1+α,∞ if ∂y a ∈ Hα,∞ , in which case M1+α,∞ (a) is deﬁned as Mα,∞ (∂y a). Finally, Hα is the set of all continuous functions f (x) on [0, 1] for which |f (x2 ) − f (x1 )| Mα (f ) := sup < ∞, |x2 − x1 |α x1 ,x2 and H1+α is the space of all continuously diﬀerentiable functions f (x) on [0, 1] with M1+α (f ) := Mα (f ) < ∞; we put M∞ (f ) = supx∈[0,1] |f (x)|.

3. Counterexamples In this section we prove Theorem 1.1 and the ﬁrst statement of Theorem 1.4. We show that counterexamples can even be found within the functions a(x, y, t) that are independent of one of the variables x and y. Theorem 3.1. There exist functions a(x, t) in C([0, 1] × T) such that sup AN (a)∞ = ∞.

N ≥0

Proof. Assume the contrary, that is, sup AN (a)∞ < ∞ for every function a in C([0, 1]× T). Let S denote the Banach space of all sequences {BN }∞ N =0 of matrices (N +1)×(N +1) such that BN ∈ C {BN }∞ N =0 := sup BN ∞ < ∞. N ≥0

By our assumption, the map T : C([0, 1] × T) → S,

a → {AN (a)}∞ N =0

is a linear operator deﬁned on all of C([0, 1] × T). To show that T is bounded, we employ the closed graph theorem. Thus, let an , a ∈ C([0, 1] × T) and suppose an → a in C([0, 1] × T) and T an → b = {BN }∞ N =0 in S. Then, for ﬁxed N ≥ 0, AN (an ) − BN ∞ → 0 as n → ∞ and hence the jk entry of AN (an ) converges to

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the jk entry of BN as n → ∞, that is, [AN (an )]jk → [BN ]jk for 0 ≤ j, k ≤ N . On the other hand, 1 1 1 j j 11 | [AN (an )]jk − [AN (a)]jk | = 11 (ˆ an )j−k −a ˆj−k N N 1 1 14

1 |dt| 11 j j an ,t − a ,t t−(j−k) ≤ M∞,∞ (an − a) = o(1), = 11 N N 2π 1 T which yields the equality [BN ]jk = [AN (a)]jk . Consequently, T a = b. The closed graph theorem therefore implies that T is bounded. We have shown that there is a constant C < ∞ such that AN (a)∞ ≤ CM∞,∞ (a)

(6)

for all a ∈ C([0, 1] × T). Fix N ≥ 2 and for j = 1, . . . , N − 1, denote by Ij the segment

1 j 1 j − , + . Ij = N 2N N 2N Let aj be the function that is identically zero on [0, 1] \ Ij , increases linearly from 0 to 1 on the left half of Ij , and decreases linearly from 1 to 0 on the right half of Ij . Put a(x, t) = a1 (x)t1 + a2 (x)t2 + · · · + aN −1 (x)tN −1 . As the spectral norm of a matrix is greater than or equal to the 2 norm of its ﬁrst column and as a ˆj (x) = aj (x) for 1 ≤ j ≤ N − 1, it follows that 12 N N −1 1 −1 1 j 11 2 1 AN (a)∞ ≥ = 12 = N − 1. a j 1 1 N j=1 j=1 Since a(x, t) = 0 for x ∈ / ∪Ij and |a(x, t)| = |aj (x)tj | ≤ 1 for x ∈ Ij , we obtain 2 that M∞,∞ (a) = 1. Consequently, (6) gives N − 1 ≤ C 2 · 1 for all N ≥ 2, which is impossible. Theorem 3.2. If 0 < α < 1/2, there exist functions a(x, t) in Hα,∞ such that sup AN (a)∞ = ∞.

N ≥0

Proof. Assume that sup AN (a)∞ < ∞ for every a ∈ Hα,∞ . The space Hα,∞ is a Banach space under the norm a := M∞,∞ (a) + Mα,∞ (a) and hence the same argument as in the proof of Theorem 3.1 gives AN (a)∞ ≤ C(M∞,∞ (a) + Mα,∞ (a))

(7)

for all a ∈ Hα,∞ √ . Let a(x, t) be exactly as the proof of Theorem 3.1. We then have α AN (a)∞ ≥ N − √ 1, M∞,∞ (a) =α 1, and it is easily seen that Mα,∞ (a) = O(N ). Thus, (7) delivers N − 1 = O(N ), which is impossible for α < 1/2. Since AN (a(y, t)) is the transpose of AN (a(x, 1/t)), the above two theorems also deliver counterexamples with functions of the form a(y, t).

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4. Suﬃcient conditions In this section we prove the second half of Theorem 1.4 and thus also Theorem 1.2. The following result is well known; see, for example, [15, Chap. 2, Sec. 4]. We cite it with a full proof for the reader’s convenience. Lemma 4.1. If f (x) is a function in H1+α and f (0) = f (1), then f (x) =

∞

fn e2πinx

n=−∞

with |fn | ≤

Mα (f ) 2+α 2 π|n|1+α

Proof. Let |n| ≥ 1. Then 4 1 4 f (x)e−2πinx dx = fn = 0

=

1

0

f (x)d

for

|n| ≥ 1.

e−2πinx −2πin

11 4 1 4 1 1 1 e−2πinx 11 −2πinx + f (x)e dx = f (x)e−2πinx dx, f (x) −2πin 10 2πin 0 2πin 0

the last equality resulting from the requirement that f (0) = f (1). The substitution x → x − 1/(2n) yields

4 1 4 1 1 e−2πinx dx, f (x)e−2πinx dx = − f x − 2n 0 0 whence

4 1 1 1 f (x) − f x − e−2πinx dx. 4πin 0 2n Taking into account that |f (x) − f (x − 1/(2n))| ≤ Mα (f )/(2n)α , we arrive at the asserted inequality. fn =

We ﬁrst consider functions a(x, y, t) that are independent of either x or y. Theorem 4.2. Let α > 0. There exists a constant C(α) depending only on α such that AN (a)∞ ≤ C(α)(M∞,∞ (a) + M1+α,∞ (a)) for all functions a(x, t) in H1+α,∞ . Proof. We write a = a0 + a1 with a1 (x, t) = (a(1, t) − a(0, t))x + a(0, t),

a0 (x, t) = a(x, t) − a1 (x, t).

Then AN (a) = AN (a0 ) + AN (a1 ). Obviously,

N j AN (b(x)c(t)) = b cˆj−k = DN (b)TN (c), N j,k=0

(8)

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cj−k )N where DN (b) = diag(b(j/N ))N j=0 and TN (c) = (ˆ j,k=0 . Taking into account that DN (b)∞ ≤ M∞ (b) and TN (c)∞ ≤ M∞ (c), we obtain that AN (a1 )∞ ≤ M∞ (x)M∞ (a(1, t) − a(0, t)) + M∞ (a(0, t)) ≤ 3M∞,∞ (a). As a0 (0, t) = a0 (1, t) (= 0), Lemma 4.1 gives ∞

a0 (x, t) =

a0n (t)e2πinx

n=−∞

with |a0n (t)| ≤

Mα (∂x a0 (x, t)) 22+α π|n|1+α

(9)

for |n| ≥ 1. From (8) we infer that AN (a0n (t)e2πinx )∞ ≤ M∞ (e2πinx )M∞ (a0n (t)) = M∞ (a0n ). Thus, by (9), AN (a0 )∞

≤ M∞ (a00 ) +

M∞ (a0n )

|n|≥1

≤ M∞ (a00 ) +

Mα,∞ (∂x a0 (x, t)) . 22+α π |n|1+α 1

|n|≥1

Since a0 (x, t) = a(x, t) − a1 (x, t) and ∂x a1 (x, t) is independent of x, we get Mα,∞ (∂x a0 (x, t)) = Mα,∞ (∂x a(x, t)) = M1+α,∞ (a). Furthermore, M∞ (a00 )

1 1 a0 (x, t)dx11 ≤ M∞,∞ (a0 ) = M∞,∞ (a − a1 )

=

14 1 sup 11 t∈T

≤

M∞,∞ (a) + M∞,∞ (a1 ) ≤ 4 M∞,∞ (a).

0

1

In summary,

⎛

AN (a)∞ ≤ 7 M∞,∞ (a) + ⎝

1 22+α π

|n|≥1

⎞ 1 ⎠ M1+α,∞ (a), |n|1+α

which implies the assertion at once.

Theorem 4.3. Let α > 0. There exists a constant C(α) that depends only on α such that AN (a)∞ ≤ C(α)(M∞,∞ (a) + M1+α,∞ (a)) for all functions a(y, t) in H1+α,∞ . Proof. This follows from Theorem 4.2 by taking transposed matrices.

We now turn to the case where a(x, y, t) depends on all of the three variables.

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Lemma 4.4. Let f (x, y, t) be a function in H1+α,1+α,∞ (α > 0) and suppose f (0, y, t) = f (1, y, t) for all y and t. Then ∞ f (x, y, t) = fn (y, t)e2πinx n=−∞

with 1 M1+α,∞,∞(f ), 22+α π|n|1+α 1 M1+α,1+α,∞(f ) M1+α,∞ (fn ) ≤ 2+α 2 π|n|1+α M∞,∞ (fn ) ≤

(10) (11)

for |n| ≥ 1. Proof. Estimate (10) is immediate from Lemma 4.1 and the deﬁnition of the number M1+α,∞,∞ (f ). To prove (11), we ﬁrst note that 4 1 (∂y f (x, y2 , t) − ∂y f (x, y1 , t))e−2πinx dx (12) ∂y fn (y2 , t) − ∂y fn (y1 , t) = 0

and ∂y f (0, y, t) = ∂y (1, y, t). Integrating by parts we therefore see that (12) equals 4 1 1 (∂x ∂y f (x, y2 , t) − ∂x ∂y f (x, y1 , t))e−2πinx dx, 2πin 0 and the substitution x → x − 1/(2n) shows that this is

4 1 1 1 , x, y1 , y2 , t e−2πinx dx. Δ2 ∂x ∂y f x + 4πin 0 2n Consequently, 4 1 1 1 |∂y fn (y2 , t) − ∂y fn (y1 , t)| ≤ M1+α,1+α,∞ (f ) |y2 − y1 |α dx, 4π|n| 0 |2n|α

which gives (11).

Theorem 4.5. Let α > 0. Then there exists a constant D(α) < ∞ depending only on α such that AN (a)∞ ≤ D(α)(M∞,∞,∞ (a)+M1+α,∞,∞ (a)+M∞,1+α,∞ (a)+M1+α,1+α,∞ (a)) for all N ≥ 0 and all functions a(x, y, t) in H1+α,1+α,∞ . Proof. We write a = a0 + a1 and accordingly AN (a) = AN (a0 ) + AN (a1 ) with a1 (x, y, t) = (a(1, y, t) − a(0, y, t))x + a(0, y, t), a0 (x, y, t) = a(x, y, t) − a1 (x, y, t). For every function c(y, t) we have N N k j j AN (c(y, t)x) = cˆj−k = diag AN (c(y, t)) N N j,k=0 N j=0

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and the spectral norm of the diagonal matrix is 1. Hence AN (a1 )∞

≤

AN (a(1, y, t) − a(0, y, t))∞ + AN (a(0, y, t))∞

≤

AN (a(1, y, t))∞ + 2 AN (a(0, y, t))∞ .

Theorem 4.3 gives AN (a(1, y, t))∞

≤ ≤

C(α)(M∞,∞ (a(1, y, t)) + M1+α,∞ (a(1, y, t))) C(α)(M∞,∞,∞ (a) + M∞,1+α,∞ (a)),

AN (a(0, y, t))∞

≤ ≤

C(α)(M∞,∞ (a(0, y, t)) + M1+α,∞ (a(0, y, t))) C(α)(M∞,∞,∞ (a) + M∞,1+α,∞ (a)).

Thus, AN (a1 )∞ ≤ 3 C(α)(M∞,∞,∞ (a) + M∞,1+α,∞ (a)). On the other hand, since a0 (0, y, t) = a0 (1, y, t), Lemma 4.4 implies that a0 (x, y, t) =

∞

a0n (y, t)e2πinx

n=−∞

where the functions

a0n (y, t)

satisfy estimates like (10) and (11). This time

N k e2πinj/N (ˆ a0n )j−k AN (a0n (y, t)e2πinx ) = N j,k=0 N = diag e2πijn/N AN (a0n (y, t)) j=0

and the spectral norm of the diagonal matrix is again 1. It follows from Theorem 4.3 and Lemma 4.4 that AN (a0 )∞ ≤ AN (a00 )∞ + AN (a0n )∞ ≤ AN (a00 )∞ +

|n|≥1

|n|≥1

1 (M1+α,∞,∞ (a0 ) + M1+α,1+α,∞ (a0 )) 22+α π|n|1+α

=: AN (a00 )∞ + D0 (α)(M1+α,∞,∞ (a0 ) + M1+α,1+α,∞ (a0 )) We have M∗ (a0 ) = M∗ (a − a1 ) ≤ M∗ (a) + M∗ (a1 ). Since a1 (x, y, t) depends on x linearly, we get M1+α,∞,∞ (a1 ) = M1+α,1+α,∞ (a1 ) = 0. Thus, M1+α,∞,∞ (a0 ) + M1+α,1+α,∞ (a0 ) ≤ M1+α,∞,∞ (a) + M1+α,1+α,∞ (a) Finally, Theorem 4.3 shows that

As a00 (t) =

31 0

AN (a00 )∞ ≤ C(α)(M∞,∞ (a00 ) + M1+α,∞ (a00 )). a0 (x, y, t)dx, it results that

M∞,∞ (a00 )

≤ M∞,∞,∞ (a0 ) = M∞,∞,∞ (a − a1 ) ≤ M∞,∞,∞ (a) + M∞,∞,∞ (a1 ) ≤ 4 M∞,∞,∞ (a0 )

Toeplitz Matrices with Variable Coeﬃcients and

4

≤

|∂y a0 (x, y2 , t) − ∂y a0 (x, y1 , t)| dx |y2 − y1 |α t∈T y1 ,y2 0 M∞,1+α,∞ (a0 ) ≤ M∞,1+α,∞ (a) + M∞,1+α,∞(a1 )

=

M∞,1+α,∞ (a) + 3 M∞,1+α,∞ (a).

M1+α,∞ (a00 ) ≤

201

1

sup sup

Putting things together we arrive at the theorem with D(α) = 7 C(α)+D0 (α).

Clearly, Theorem 4.5 implies the second half of Theorem 1.4 and thus also Theorem 1.2.

5. Discontinuous generating functions So far we have assumed that a ∈ C([0, 1]2 × T). Inequality (4) implies that sup AN (a)∞ <∞ if a is given by (1) and a ˆn are any bounded functions on [0, 1]2 such that ∞ M (ˆ a ) < ∞. Thus, suﬃcient smoothness in t allows ∞,∞ n n=−∞ us to admit arbitrary bounded coeﬃcient a ˆn (x, y). For 0 < α ≤ 1, we denote by H1+α,1+α the Banach space of all continuous functions f : [0, 1]2 → C which have continuous partial derivatives up to the order 2 and for which f 1+α := M∞,∞ (f ) + Mα,∞ (∂x f ) + M∞,α (∂y f ) + Mα,α (∂x ∂y f ) < ∞, where M∞,∞ (f ) is the maximum modulus of f (x, y) on [0, 1]2 and Mα,∞ (g) = sup

sup

y∈[0,1] x1 ,x2

M∞,α (g) = sup sup

x∈[0,1] y1 ,y2

Mα,α (g) = sup sup x1 ,x2 y1 ,y2

|g(x2 , y) − g(x1 , y)| , |x2 − x1 |α |g(x, y2 ) − g(x, y1 )| , |y2 − y1 |α

|Δ2 g(x1 , x2 , y1 , y2 )| . |x2 − x1 |α |y2 − y1 |α

Let L∞ (T, H1+α,1+α ) be the set of all measurable and essentially bounded functions a : T → H1+α,1+α . A check of the proofs shows that these work literally also for functions a in L∞ (T, H1+α,1+α ). Thus, if a is in L∞ (T, H1+α,1+α ) then sup AN (a)∞ < ∞. Since C 4 ([0, 1]2 ) ⊂ H2,2 , we obtain in particular Theorem 1.3.

References [1] D. Borthwick and A. Uribe, On the pseudospectra of Berezin-Toeplitz operators, Methods Appl. Anal. 10 (2003), 31–65. [2] A. B¨ ottcher and S. Grudsky, Uniform boundedness of Toeplitz matrices with variable coeﬃcients, Integral Equations Operator Theory 60 (2008), 313–328. [3] T. Ehrhardt and B. Shao, Asymptotic behavior of variable-coeﬃcient Toeplitz determinants, J. Fourier Anal. Appl. 7 (2001), 71–92.

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[4] D. Fasino and S. Serra Capizzano, From Toeplitz matrix sequences to zero distribution of orthogonal polynomials, Fast Algorithms for Structured Matrices: Theory and Applications (South Hadley, MA, 2001), 329–339, Contemp. Math., 323, Amer. Math. Soc., Providence, RI 2003. [5] M. Kac, W.L. Murdock, and G. Szeg¨ o, On the eigenvalues of certain Hermitian forms, J. Rational Mech. Anal. 2 (1953), 767–800. [6] A.B. Kuijlaars and S. Serra Capizzano, Asymptotic zero distribution of orthogonal polynomials with discontinuously varying recurrence coeﬃcients, J. Approx. Theory 113 (2001), 142–155. [7] V. Rabinovich, S. Roch, and B. Silbermann, Limit Operators and Their Applications in Operator Theory, Operator Theory: Advances and Applications, 150, Birkh¨ auser Verlag, Basel 2004. [8] S. Serra Capizzano, Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial diﬀerential equations, Linear Algebra Appl. 366 (2003), 371–402. [9] B. Shao, On the singular values of generalized Toeplitz matrices, Integral Equations Operator Theory 49 (2004), 239–254 [10] I.B. Simonenko, Szeg¨ o-type limit theorems for generalized discrete convolution operators (Russian), Mat. Zametki 78 (2005), 265–277. [11] P. Tilli, Locally Toeplitz sequences: spectral properties and applications, Linear Algebra Appl. 278 (1998), 91–120. [12] L.N. Trefethen and S.J. Chapman, Wave packet pseudomodes of twisted Toeplitz matrices, Comm. Pure Appl. Math. 57 (2004), 1233–1264. [13] O.N. Zabroda, Generalized Convolution Operators and Asymptotic Spectral Theory, Dissertation, TU Chemnitz 2006. [14] O.N. Zabroda and I.B. Simonenko, Asymptotic invertibility and the collective asymptotic behavior of the spectrum of generalized one-dimensional discrete convolutions, Funct. Anal. Appl. 38 (2004), 65–66. [15] A. Zygmund, Trigonometric series, 2nd ed., Vols. I, II, Cambridge University Press, New York 1959. Albrecht B¨ ottcher Fakult¨ at f¨ ur Mathematik TU Chemnitz D-09107 Chemnitz, Germany e-mail: [email protected] Sergei Grudsky Departamento de Matem´ aticas CINVESTAV del I.P.N. Apartado Postal 14-740 07000 M´exico, D.F., M´exico e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 199, 203–220 c 2010 Birkh¨ auser Verlag Basel/Switzerland

A Priori Estimates on the Structured Conditioning of Cauchy and Vandermonde Matrices Enrico Bozzo and Dario Fasino Abstract. We analyze the componentwise and normwise sensitivity of inverses of Cauchy, Vandermonde, and Cauchy-Vandermonde matrices, with respect to relative componentwise perturbations in the nodes deﬁning these matrices. We obtain a priori, easily computable upper bounds for these condition numbers. In particular, we improve known estimates for Vandermonde matrices with generic real nodes; we consider in detail Vandermonde matrices with nonnegative or symmetric nodes; and we extend the analysis to the class of complex Cauchy-Vandermonde matrices. Mathematics Subject Classiﬁcation (2000). Primary 15A12; Secondary 15A57, 65F35. Keywords. Condition number, displacement structure, Cauchy matrix, Vandermonde matrix.

1. Introduction A structured matrix is, in some sense, a matrix whose entries depend on a small set of parameters. Clearly, this dependence is somehow preserved by matrix inversion. The main question addressed here is: How sensitive is the inverse of a structured matrix to perturbations in its parameters? The answer to the above question is what we call structured conditioning. This paper focuses on the structured conditioning of Cauchy, Vandermonde and CauchyVandermonde matrices, which have been one of the main research topics of Georg Heinig, see, e.g., [9, 12, 13]. The main reasons motivating this study are the analysis of the inﬂuence of data errors in the solution of linear systems with structured matrices, and the assessment of stability properties of fast algorithms for solving such linear systems. These algorithms act directly on the set of parameters deﬁning a structured matrix rather than on its entries, see, e.g., [3, 4, 5, 9, 10, 12, 15], hence

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their stability characteristics, which are sometimes surprising, should be examined in the light of suitable reﬁnements of classical condition numbers, taking in due consideration both more accurate measures of perturbations and the structure of the problem considered, as observed, e.g., in [1, 10, 11, 14, 15, 22]. Indeed, suppose that the solution computed by such an algorithm in ﬁnite precision is the exact solution of a similarly structured system deﬁned by slightly perturbed parameters. To assess the quality of the computed solution with no further assumptions on the right-hand side, we should make use of some measure of the sensitivity of matrix inversion with respect to perturbation in its parameters. Theoretical bases of this argument can be found in [14], where such a backward structured error analysis in introduced rather formally for structured matrices whose dependence on the parameters is linear, and in [1, 15, 22], where a detailed analysis has been carried out for Vandermonde systems, mainly motivated by the stability analysis of the Bj¨ ork-Pereyra algorithm. Other results in the same streamline can be found in [7, 25], concerning possibly rectangular Cauchy, Vandermonde, Toeplitz and Hankel matrices. By the way, the results in the above-mentioned papers are mainly aimed at characterizing the structured conditioning in terms of exact (possibly generalized) inverse matrices and solutions of suitable linear systems, hence they are a posteriori results, and the resulting expressions may be hard to compute. The goal of this paper is to obtain a priori bounds for the structured conditioning of Cauchy, Vandermonde and other related matrices, that are easily computable right from their parameters, with no involvement of exact (generalized) inverse matrices. Besides to their pervasive occurrence in computations with polynomial and rational functions, Cauchy and Vandermonde matrices play an important role in deriving structural and computational properties of many relevant matrix classes with displacement structure, see, e.g., [9, 13, 17, 18, 19]. For example, they occur as fundamental blocks (together with trigonometric transforms) in decomposition formulas for Toeplitz, Hankel, and related matrices. In this paper, we will pay particular attention to the structured conditioning of the individual columns of their inverses, as they have a special relevance in polynomial and rational interpolation problems. Indeed, let x1 . . . xn be pairwise distinct points in the complex plane, considered as parameters, and let φ1 (x) . . . φn (x) be a ﬁxed set of functions such that the collocation matrix X ≡ (φj (xi )) is nonsingular. In fact, Vandermonde, Cauchy, and Cauchy-Vandermonde matrices arise as collocation matrices when the functions φj (x) are monomials or particular rational functions. Let X −1 ≡ (vi,j ). Then, the functions k (x) = j vj,k φj (x) for 1 ≤ k ≤ n are the Lagrange functions for the interpolation problem deﬁned by the points xi and the functions φj (x). Hence, a further reason for investigating the structured conditioning of X −1 is to give a measure of the sensitivity of the functions k (x) with respect to perturbations in the interpolation points xi . After giving in Section 2 a quick look at Cauchy matrices, we consider Vandermonde matrices in Section 3. There, we will improve the result in [11, Thm. 1] on the mixed structured conditioning of Vandermonde matrices (see Corollary 2),

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205

and consider in detail Vandermonde matrices with nonnegative nodes and symmetric nodes. In Section 4 we will extend our analysis to Cauchy-Vandermonde matrices with complex nodes. 1.1. Main deﬁnitions and notations We borrow from [10, 11] the following deﬁnitions and notations. Let p and q be positive integers, and let F be a (densely deﬁned) continuous function F : Cp → Cq . One usually deﬁnes the normwise condition number of F in a given point x ∈ Cp as F (x) − F (˜ x) x . κ(F, x) = lim sup ε→0 x−˜ F (x) x − x ˜ x≤ε In the limit as the perturbation size tends to zero, this number gives the worst possible magniﬁcation of the quantity x − x˜/x in the computation of F . Here and in what follows, · = · ∞ , unless otherwise noted. The number x − x ˜/x is the relative normwise distance between x and x˜. In what follows, we also consider the relative componentwise distance between two points x, x ˜ ∈ Cp , deﬁned as δ(x, x ˜) = min{ε : |xi − x ˜i | ≤ ε|xi |, 1 ≤ i ≤ p}.

(1)

The above deﬁnition is of interest in numerical analysis, because it is the most appropriate way to measure errors induced by the ﬁnite precision representation of machine numbers. Observe that if an entry of the vector x is zero, the corresponding entry in x˜ must be zero for δ(x, x ˜) be deﬁnite: Componentwise relative perturbations do not aﬀect null entries. Accordingly, we consider the componentwise condition number of F in x = 0 as δ(F (x), F (˜ x)) c(F, x) = lim sup . (2) ε→0 δ(x,˜ δ(x, x ˜) x)≤ε Throughout this paper, we consider matrix and vector moduli and inequalities as applied componentwise. Hence, we observe that c(F, x) is characterized by the following inequality: |F (x) − F (˜ x)| ≤ |F (x)|c(F, x)δ(x, x˜) + o(δ(x, x˜)).

(3)

In some cases, it may be also of interest to consider the mixed condition number m(F, x) = lim

sup

ε→0 δ(x,˜ x)≤ε

F (x) − F (˜ x) 1 . F (x) δ(x, x ˜)

(4)

Since x − x ˜/x ≤ δ(x, x ˜), we have both m(F, x) ≤ c(F, x) and m(F, x) ≤ κ(F, x), but in general c(F, x) and κ(F, x) are unrelated. Moreover, the equivalent deﬁnition D(x − x ˜) δ(x, x ˜) = sup Dx D diagonal implies that m(F, x) results by minimizing κ(F, x) with respect to all argument normalizations, while c(F, x) = supD m(DF, x) is a “worst case” measure of the

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sensitivity of F in presence of diagonal scalings in both the parameters and the function values. If the map F is diﬀerentiable, the condition numbers introduced above can be related to its diﬀerential as follows: For any n-vector a = (a1 , . . . , an )T , let Da denote the n × n diagonal matrix whose ith diagonal entry is ai . Then, if F denotes the diﬀerential of F , we have κ(F, x)

= F (x)x/F (x)

(5)

c(F, x)

=

(6)

m(F, x)

=

DF−1(x) F (x)Dx F (x)Dx /F (x).

(7)

Some further notations are used throughout this paper: Let ei be the ith standard basis vector, whose order will be made clear from the context, and 1 = 2 (1, . . . , 1)T . Let Vec : Cn×n → Cn be the operator such that Vec(X) is the n2 order vector obtained by stacking downward the columns of X, into one long column. Moreover, for x = (x1 , . . . , xp )T ∈ Cp and y = (y1 , . . . , yq )T ∈ Cq , let Δ(x) = max i =j

|xi | , |xi − xj |

Δ(x, y) = max i,j

|xi | . |xi − yj |

(8)

In the above equations, fractions with vanishing denominators assume the value +∞, whatever the numerators are. In particular, Δ(x) < +∞ if and only if the points x1 . . . xn are pairwise distinct, and similarly for Δ(x, y). 1.2. Basic displacement structured matrices Cauchy, Vandermonde and Cauchy-Vandermonde matrices are among the best known matrices with a displacement structure. Such kind of structure is deﬁned in terms of a displacement operator LM,N (X) = M X − XN, where M, N are two ﬁxed n×n matrices with disjoint spectra, so that the operator LM,N is invertible. Throughout this paper, we will deal with matrix spaces having the following form: DM,N = {X : rank(LM,N (X)) = 1}. A basic fact that will be used here to reduce the analysis of Vandermonde and Cauchy-Vandemonde matrices to the simpler case of Cauchy matrices is this: If N = SDS −1 and X ∈ DM,N , then XS ∈ DM,D . Regarding the connection between displacement structure and conditioning, we recall that in the paper [20] an exponentially growing lower bound is derived for the spectral conditioning of matrices X such that AX + XAT = −BB T , where B has low rank and all eigenvalues of A have negative real part. By Lyapunov theorem, any such matrix X is symmetric and positive deﬁnite; remarkably, the displacement structure induced by the operator LA,−AT forces X to be very badly conditioned.

Structured Conditioning of Cauchy and Vandermonde Matrices

207

2. Cauchy matrices Let x = (x1 , . . . , xn )T ∈ Cn and y = (y1 , . . . , yn )T ∈ Cn have pairwise distinct entries, with xi = yj for 1 ≤ i, j ≤ n. The Cauchy matrix associated with x and y is deﬁned by Cx,y ≡ (1/(xi − yj )). Since the displacement operator LDx ,Dy is nonsingular, Cx,y is the unique solution of the displacement equation Dx Cx,y − Cx,y Dy = 11T . The following explicit formula for the entries of the inverse of Cx,y −1 is well known, see, e.g., [10]: For Cx,y ≡ (vi,j ) we have vi,j =

− xl ) 1 (y − y ) y − xj i l i l =i l (yi

− yl ) . (x − xl ) j l =j l (xj

(9)

Remark that vi,j = 0. In [10] the above expression is diﬀerentiated with respect to xi and yi in order to bound the componentwise conditioning of the inversion of Cx,y , via (6). In the next result we show a simple bound for the −1 componentwise conditioning of the columns of Cx,y , when only the x node vector is subject to perturbations. This result plays a fundamental role in the subsequent sections. Moreover, as mentioned in the Introduction, it is useful to estimate the sensitivity of a set of rational Lagrange functions with respect to the interpolation nodes. It is apparent from the deﬁnition of Cx,y that we obtain the corresponding result for y by simply considering the transpose matrix. Theorem 1. Let 1 ≤ i ≤ n and y ∈ Cn be ﬁxed, with pairwise distinct entries. Let −1 Fi : Cn → Cn be deﬁned as Fi (x) = Cx,y ei . Then, c(Fi , x) ≤ 2(n − 1)Δ(x) + (2n − 1)Δ(x, y). Proof. From (6) we have

1 1 1 1 1 xk ∂vj,i 1 1 1 1 1 = max 1xk ∂ log |vj,i |1 , 1 1 1 1 1≤j≤n 1≤j≤n vj,i ∂xk ∂xk

c(Fi , x) = max

k

(10)

k

where vj,i is given in (9). For k = i we have: 1 1 1 1 1 ∂ log |vj,i | 1 1 xk 11 1 = 1− xk 1xk ≤ Δ(x) + Δ(x, y). 1 ∂xk 1 1 xk − xi xk − yj 1 On the other hand, 1 1 1 1 1 1 xi 1 1 ∂ log |vj,i | 1 1 xi 1xi 1 ≤ (n − 1)Δ(x) + nΔ(x, y). 1=1 − 1 ∂xi 1 11 xi − xl xi − yl 11 l =i l Plugging the latter inequalities into (10) we arrive at the claim.

For notational simplicity, in the sequel we use the shorthand Δ(n, x, y) = 2(n − 1)Δ(x) + (2n − 1)Δ(x, y).

(11)

In the next corollary we consider the structured conditioning of the matrices in DDx ,Dy . Observe that any matrix in this set can be expressed as D1 Cx,y D2 for some diagonal matrices D1 and D2 .

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−1 D2 ), where Corollary 1. Let F : Cn → Cn be deﬁned as F (x) = Vec(D1 Cx,y D1 , D2 are arbitrary diagonal matrices and y is as in the preceding theorem. Then, 2

c(F, x) ≤ Δ(n, x, y), where Δ(n, x, y) is deﬁned in (11). −1 Proof. Scaling the entries of Cx,y by constant factors does not aﬀect their conditioning, as it should be clear from the deﬁnitions (1) and (2). Hence we can assume Dc = Dd = I. In this case, we observe that

c(F, x) = max c(Fi , x) ≤ Δ(n, x, y), 1≤i≤n

where Fi is as in the preceding theorem.

If F is as in the preceding corollary and Dc = Dd = I, then F (x) is the map −1 −1 z → Vec(Cx,y E(z)Cx,y ), where E(z) ≡ (−zi /(xi − yj )2 ). Hence, in view of (5), the normwise conditioning κ(F, x) is essentially driven by the (unstructured) condi−1 tioning of Cx,y , that is, Cx,y Cx,y . Quite few estimates are currently available for the latter, and only for special vectors x and y. Here we only mention the paper [23], dealing with the spectral conditioning of the Cauchy-Toeplitz matrix CT ≡ (1/(a+i−j)), and [8], for Cauchy matrices C−y,y with positive vector y (note that these two examples fall into the generic case considered in [20] and mentioned in Subsection 1.2). Generally, all these matrices are very ill conditioned, and this fact should be contrasted with the slowly-growing estimate for the componentwise conditioning shown in the above corollary; for example, if y has positive entries we have (see [8])

2n−2 yn + y 1 −1 C−y,y 2 C−y,y 2 > . y n − y1 Note that also the exponentially ill-conditioned Hilbert matrix CH ≡ (1/(i+j −1)) is a special Cauchy matrix. For that matrix, an O(n2 ) bound for its structured conditioning is derived in [10], while its spectral conditioning grows roughly like 34n [2].

3. Vandermonde matrices Given a vector x = (x1 , . . . , xn )T ∈ Cn with pairwise distinct entries, for any ) fulﬁlls the displacement ﬁxed 0 ≤ θ < 2π the Vandermonde matrix Vx ≡ (xj−1 i equation ⎛ n ⎞ x1 − einθ ⎜ ⎟ T .. LDx ,Pθ (Vx ) = Dx Vx − Vx Pθ = ⎝ ⎠ en , . xnn − einθ

Structured Conditioning of Cauchy and Vandermonde Matrices where i is the imaginary unit, and ⎛

0

⎜ ⎜ 1 Pθ = ⎜ ⎜ ⎝ O

··· .. . .. .

0

einθ

0 .. .

0 .. .

1

0

209

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

Recall that the spectral decomposition of Pθ is explicitly computable, Pθ = eiθ D(θ)ΦDξ Φ−1 D(θ)−1 , (12) 5 where ξ = (1, ei2π/n , . . . , ei(n−1)2π/n )T , Φ = 1/n Vξ is the unitary Fourier matrix of order n, and D(θ) = Diag(1, e−iθ , . . . , e−i(n−1)θ ). Only for notational simplicity, in what follows we suppose that the operator LDx ,P0 is nonsingular, that is, Δ(x, ξ) < +∞. This hypothesis is of no restriction, and can always be fulﬁlled by a suitable rotation of the complex plane. Indeed, let x ˆ = e−iθ x; we have Vx D(θ) = Vxˆ and, from (12), LDx ,Pθ (Vx )

= [Dx Vx D(θ) − eiθ Vx D(θ)P0 ]D(θ)−1 = eiθ [Dxˆ Vxˆ − Vxˆ P0 ]D(θ)−1 = einθ LDxˆ ,P0 (Vxˆ ).

The last equation uses the fact that eTn D(θ)−1 = ei(n−1)θ eTn . Hence, the discussion of the Vandermonde matrix with node vector x, for a chosen matrix Pθ , can be carried out equivalently by considering the Vandermonde matrix deﬁned by xˆ and the displacement equation with the matrix P0 . Since the entries of Vxˆ and Vxˆ−1 have the same modulus of the entries of Vx and Vx−1 , respectively, the rotation x → x ˆ leaves unaltered the conditioning properties we are investigating. In contrast to what happens with Cauchy matrices, inverse Vandermonde matrices can have zero entries, hence the componentwise conditioning of Vandermonde matrices cannot be bounded in general. For this reason, we will consider the mixed conditioning, in the generic case, as in [11, 15, 22]. Furthermore, we will obtain estimates for the componentwise conditioning for particular conﬁgurations of the nodes, namely, nonnegative or symmetric nodes. We will use the following lemma to bridge Cauchy and Vandermode matrices. The statement can be obtained as a consequence of Proposition 3.2 in [9]. For convenience, we provide here a short and self-contained proof. Lemma 1. Let x = (x1 , . . . , xn )T ∈ Cn and a = (xn1 − 1, . . . , xnn − 1)T . If Da is nonsingular, the matrix Vx can be factorized as follows: 1 Vx = √ Da Cx,ξ Dξ−1 Φ−1 . n Proof. From (12) we have 1 Dx Vx Φ − Vx ΦDξ = (Dx Vx − Vx P0 )Φ = aeTn Φ = √ a1T Dξ−1 . n

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Since diagonal matrices commute, we have LDx ,Dξ (Da−1 Vx ΦDξ ) = = =

Da−1 [Dx Vx Φ − Vx ΦDξ ]Dξ 5 1/n Da−1 a1T Dξ−1 Dξ 5 1/n 11T .

By hypothesis, the nodes xi are not roots of unit, hence the operator LDx ,Dξ is 5 invertible. Thus Da−1 Vx ΦDξ = 1/n Cx,ξ , whence we obtain the thesis. Observe that, if the hypothesis on Da is false, then the matrix Cx,ξ is not even deﬁned. Indeed, the same hypothesis can be restated as Δ(x, ξ) < +∞. √ Lemma 2. Let F : Cn → Cn be the map F (v) = Φv. Then, m(F, v) = n.

Proof. See [6, Corollary 1].

Theorem 2. Let 1 ≤ i ≤ n be ﬁxed, and let Fi : Cn → Cn be the function deﬁned as Fi (x) = Vx−1 ei . Furthermore, let ξ = (1, ei2π/n , . . . , ei(n−1)2π/n )T . Then, for any 0 ≤ θ < 2π we have 1 1 1 nxni 1 √ −iθ 1 , 1 m(Fi , x) ≤ n Δ(n, e x, ξ) + 1 n xi − einθ 1 where Δ(n, e−iθ x, ξ) can be obtained from (11). Proof. In the light of the argument outlined at the beginning of this section, we can restrict the proof to the case θ = 0, since the general case follows √ by considering −1 −1 Da , the matrix Vxˆ , with xˆ = e−iθ x. From Lemma 1 we obtain Vx−1 = nΦDξ Cx,ξ n n T where a = (x1 − 1, . . . , xn − 1) . Hence, for 1 ≤ i ≤ n, √ n −1 Fi (x) = n ΦDξ Cx,ξ ei . xi − 1 Consider the decomposition Fi (x) = G(Hi (x)), where G(x) = Φx and Hi (x) = √ −1 n(xni − 1)−1 Dξ Cx,ξ ei . It can be shown that m(Fi , x) ≤ m(G, Hi (x))c(Hi , x), see √ [11, p. 692]. Then, from Lemma 2 we obtain m(Fi , x) ≤ nc(Hi , x). Furthermore, for x and x ˜ such that δ(x, x ˜) = , we have from Theorem 1 √ n −1 x)| ≤ n|xi − 1|−1 |Cx,ξ ei − Cx˜−1 |Hi (x) − Hi (˜ ,ξ ei | √ n −1 n xi − 1)−1 ||Cx˜−1 + n|(xi − 1) − (˜ ,ξ ei | 1 1 n 1 nx 1 ≤ Δ(n, x, ξ)|Hi (x)| + 11 n i 11 |Hi (x)| + o(). x −1 i

From the equivalence of (3) and (2) we have

1 1 1 nxni 1 1 1, c(Hi , x) ≤ Δ(n, x, ξ) + 1 n x − 11 i

completing the proof.

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211

√ Note that the factor n in the right-hand side of the thesis of the preceding theorem can be dropped oﬀ, if the 2-norm is used instead of the ∞norm in the deﬁnition of m(Fi , x). Indeed, one proves easily that m2 (G, x) = G (x)Dx 2 /G(x)2 = 1, and we have the attainable upper bound m2 (Fi , x) ≤ c(Hi , x), in the notations of the preceding proof. Hence, in some sense, the Euclidean norm is more appropriate than the ∞-norm to analyze the structured conditioning of Vx . In the case of real nodes, Gohberg and Koltracht proved in [11] the upper bound m(Fi , x) ≤ n2 max(nΔ(x), n + Δ(x)). We improve this bound in the following corollary. Corollary 2. In the notations of Theorem 2, if x ∈ Rn we have √ m(Fi , x) < n(2(n − 1)Δ(x) + 2n2 ). Proof. Let θ = π/(2n). We have: Δ(x, eiθ ξ) ≤

π 1 < = n. sin(θ) 2θ

Hence from (11) Δ(n, e−iθ x, ξ) = Δ(n, x, eiθ ξ) < 2(n − 1)Δ(x) + (2n − 1)n. Moreover,

1 1 1 n 1 1 nxni 1 1 1 1 1 = n 1 xi 1 < n, 1 xn − einθ 1 1 xn − i 1 i

i

and the claim follows from Theorem 2.

We consider the complete inverse of Vx in the next corollary, which follows from an argument analogous to that used in Corollary 1. Corollary 3. Let F : Cn → Cn be deﬁned as F (x) = Vec(Vx−1 Dd ), where Dd is an arbitrary diagonal matrix whose entries do not depend on x. Then, for any 0 ≤ θ < 2π, 1 1 1 1 √ xn m(F, x) ≤ n Δ(n, e−iθ x, ξ) + n max 11 n i inθ 11 , 1≤i≤n xi − e 2

where Δ(n, e−iθ x, ξ) can be obtained from (11). Proof. Since a constant scaling of the columns of Vx−1 does not aﬀect their conditioning, we suppose Dd = I. Let Fi be the map introduced in Theorem 2 for 1 ≤ i ≤ n. Then, F (x) = (F1 (x), . . . , Fn (x))T . For arbitrary x, x ˜ we have maxi Fi (x) − Fi (x) F (x) − F (˜ x) Fi (x) − Fi (x) = ≤ max . 1≤i≤n F (x) maxi Fi (x) Fi (x) Hence, from the deﬁnition (4) we obtain m(F, x) ≤ maxi m(Fi , x), and the claim follows from Theorem 2.

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3.1. Vandermonde matrices with nonnegative nodes If xk > 0 for 1 ≤ k ≤ n then Vx−1 has no zero entries, and we can obtain precise upper bounds for its structured componentwise conditioning. Indeed, let Vx−1 ≡ (vi,j ). Then, it is well known that (n−1) vj,i = (−1)n−j σn−j (x1 , . . . , xi−1 , xi+1 , . . . , xn ) (xi − xl )−1 , l =i (n)

see, e.g., [11], where σi (a1 , . . . , an ) is the ith elementary symmetric function on n variables, (n) aj1 aj2 · · · aji . (13) σi (a1 , . . . , an ) = 1≤j1 <···<ji ≤n

By deﬁnition (13), for any 1 ≤ k ≤ n we have (n)

σi (a1 , . . . , an ) =

(n−1)

ak σi−1 (a1 , . . . , ak−1 , ak+1 , . . . , an ) (n−1)

+ σi (n)

Recall that σ0 n

(n−1)

ak σi

(n)

≡ 1 and σi

(a1 , . . . , ak−1 , ak+1 , . . . , an ).

≡ 0 for i > n or i < 0. Furthermore, we have (n)

(a1 , . . . , ak−1 , ak+1 , . . . , an ) = (i + 1)σi+1 (a1 , . . . , an ).

(14)

k=1

Actually, we can allow one entry in the vector x, say x1 , to be zero. In this case, the ﬁrst row of Vx−1 is parallel to eT1 , but no other zeros are introduced in Vx−1 ; moreover, owing to the deﬁnition (1), x1 is untouched by relative perturbations, hence the zero entries in Vx−1 don’t vary. In the light of the preceding facts, we obtain the following bound for the componentwise conditioning of the columns of Vx−1 with nodes in R+ = {x ≥ 0}: Theorem 3. Let Fi : Rn+ → Cn be the function deﬁned as Fi (x) = Vx−1 ei , for any ﬁxed 1 ≤ i ≤ n. Then c(Fi , x) ≤ (n − 1)(2Δ(x) + 1). Proof. From (6) we have

11 xk ∂vj,i 11 1 1 c(Fi , x) = max 1 vj,i ∂xk 1 . 1≤j≤n k

For k = i we have: (n−2)

xk σn−j−1 (x1 , . . . , xk−1 , xk+1 , . . . , xi−1 , xi+1 , . . . , xn ) xk ∂vj,i xk = . − (n−1) vj,i ∂xk x i − xk σ (x1 , . . . , xi−1 , xi+1 , . . . , xn ) n−j

Moreover,

xi xi ∂vj,i ∂ = xi log |vj,i | = − . vj,i ∂xi ∂xi xi − xl l =i

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213

Owing to the positivity of x1 . . . xn , from (14) we obtain 1 1 (n−1) 11 xk σn−j−1 (x1 , . . . , xk−1 , xk+1 , . . . , xi−1 , xi+1 , . . . , xn ) 11 1 1 = n − j. (n−1) 1 1 σ (x1 , . . . , xi−1 , xi+1 , . . . , xn ) k =i

n−j

Hence, we obtain

1 1 11 xk 11 11 xi 11 1, 1 1 1 c(Fi , x) ≤ max |n − j| + 1 1 xi − xk 1 + 1 1≤j≤n 1 l =i xi − xl 1 k =i

and the claim follows from the triangle inequality and the deﬁnitions (8).

The next corollary follows by an argument analogous to the ones exploited in the proof of Corollary 1 and 3, hence we omit it. Corollary 4. Let F : Rn+ → Cn be deﬁned as F (x) = Vec(Dc Vx−1 Dd ), where Dc , Dd are arbitrary diagonal matrices whose entries do not depend on x. Then, c(F, x) ≤ (n − 1)(2Δ(x) + 1). 2

The preceding results should be contrasted with well-known lower bounds on the conditioning of Vandermonde matrices with real or positive nodes [2, 16, 21, 24], which are exponentially growing functions in the order n. Moreover, we observe that the results in this subsections are trivially extended to the case where the points x1 . . . xn belong to a ray in the complex plane, xi = |xi |ω, where ω = eiθ , as it is apparent from the factorization Vx = Vω−1 x Diag(1, ω . . . ω n−1 ). 3.2. Vandermonde matrices with symmetric nodes When the nodes are restricted to be real, in many circumstances they are also symmetrically located with respect to zero. Indeed, symmetric conﬁgurations arise naturally when the nodes are Fekete points or zeros of special polynomial sequences (e.g., orthogonal polynomials from symmetric weights), or when one attempts to minimize the (classical) conditioning of Vandermonde matrices, see [16]. For the sake of simplicity, we consider the Vandermonde matrix V(x,−x) , of order 2n, whose nodes are x1 , . . . , xn , −x1 , . . . , −xn with xi > 0 (however, consider that the structured conditioning is invariant under permutation of the nodes). Introducing the vector x ˆ = (x21 , . . . , x2n )T , we have

Dxn Vn O Vx Vxˆ I I = ΠT , V(x,−x) = n O Dx Vxˆ V−x D−x V−n I −I where Π is the perfect shuﬄe permutation matrix. We obtain

−1 1 I I Vxˆ O −1 V(x,−x) = Π I −I O (Dx Vxˆ )−1 2 −1

−1 1 Vxˆ (Dx Vxˆ ) = Π , −1 −1 V −(D 2 x Vx ˆ) x ˆ

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−1 and we see from the above decomposition that every entry of the matrix V(x,−x) −1 coincides, apart of a constant, with one entry of either Vxˆ or (Dx Vxˆ )−1 . On the basis of this argument we obtain the following result: −1 Corollary 5. Let F : Rn+ → C4n be deﬁned as F (x) = Vec(Dc V(x,−x) Dd ), where Dc , Dd are diagonal matrices of order 2n, whose entries do not depend on x. Moreover, for x = (x1 , . . . , xn )T , let x ˆ = (x21 , . . . , x2n )T . Then, 2

c(F, x) ≤ 2(n − 1)(2Δ(ˆ x) + 1) + 1. Proof. We can set Dc = Dd = I, without loss in generality. Let F (1) , F (2) : Rn+ → 2 Cn be deﬁned as F (1) (x) = Vec(Vxˆ−1 ), and F (2) (x) = Vec((Dx Vxˆ )−1 ). By virtue of the preceding argument we have c(F, x) ≤ max{c(F (1) , x), c(F (2) , x)}. Our goal reduces to obtain upper bounds for c(F (1) , x) and c(F (2) , x). First, observe that F (1) (x) = G(H(x)), where G(x) = Vec(Vx−1 ) and H(x) = x ˆ. Hence c(F (1) , x) ≤ c(G, H(x))c(H, x), see [11, p. 692]. From Corollary 4 we have c(G, H(x)) ≤ (n − 1)(2Δ(ˆ x) + 1). Moreover, from (6) we obtain c(H, x) = 2. Hence c(F (1) , x) ≤ 2(n − 1)(2Δ(ˆ x) + 1). In order to estimate the componentwise conditioning of F (2) , denote Vxˆ−1 ≡ (ˆ vi,j ). Then (Dx Vxˆ )−1 ≡ (ˆ vi,j /xj ). Let ﬁx one particular pair (i, j), and consider (2) the scalar function Fi,j (x) = vˆi,j /xj , considering vˆi,j a function of x. Clearly we (2)

have c(F (2) , x) = maxi,j c(Fi,j , x). Again using (6) we obtain (2) c(Fi,j , x)

= ≤

1 1 (2) 11 xk ∂Fi,j (x) 11 1 (2) 1 1 F (x) ∂xk 1 k i,j 1 11 2

11 11 xk ∂ˆ 1 1 xj ∂ v ˆ v 1 i,j i,j 1 1 1 vˆi,j ∂xk 1 + 11 vˆi,j ∂xj xj 11 k =j

≤

c(F (1) , x) + 1.

By Corollary 4, we have c(F (2) , x) ≤ 2(n − 1)(2Δ(ˆ x) + 1) + 1 and the proof is complete. Observe that in the above corollary we have Δ(ˆ x) = max

i =j |x2 i

x2i xi xi = max < Δ(x). i =j |xi − xj | xi + xj − x2j |

Structured Conditioning of Cauchy and Vandermonde Matrices

215

4. Cauchy-Vandermonde matrices Let x = (x1 , . . . , xn )T ∈ Cn and y = (y1 , . . . , yk )T ∈ Ck , where 0 < pairwise distinct entries and xi = yj for all i, j. The n × n matrix ⎛ 1 1 · · · x1 −y 1 x1 · · · xn−k−1 x1 −y1 1 k ⎜ .. .. .. .. .. Kx,y = ⎝ . . . . . 1 1 n−k−1 · · · · · · x 1 x n n xn −y1 xn −yk

k < n, have ⎞ ⎟ ⎠

is called Cauchy-Vandermonde matrix. Matrices with the above structure appear in connection with rational interpolation problems for functions with prescribed poles, see [9, 12, 18, 19]. In particular, in [9, 18, 19] special factorizations of the inverse of Kx,y are deduced from representation formulas of the rational function solving a particular interpolation problem; such factorizations allows to compute the solution of linear systems with a Cauchy-Vandermonde matrix at the cost of O(n2 ) or even O(n log2 n) arithmetic operations. Other low-cost algorithms for such systems, based on recursions of Levinson and Schur type, are presented in [12]. For any 0 ≤ θ < 2π, the matrix Kx,y fulﬁlls the displacement equation ⎛ n−k ⎞ x1 − ei(n−k)θ

O Dy ⎜ ⎟ . T .. Nθ = , Dx Kx,y − Kx,y Nθ = ⎝ ⎠ en , e1 1T Pθ n−k i(n−k)θ xn − e where the matrix Nθ has square diagonal blocks and Pθ is as in (12). Analogously to the Vandermonde case, we can limit ourselves to consider the case θ = 0. Indeed, if we let ω = e−iθ , we have

−1 ω I O . (15) Ω= Kωx,ωy = Kx,y Ω, O Diag(1, ω, . . . , ω n−k−1 ) Since Ω is diagonal and unitary, the entries of Kωx,ωy and its inverse have the same modulus of the corresponding entries of Kx,y and its inverse, respectively, hence their structured conditioning is the same. We stress the fact that, diﬀerently to the case of Cauchy and Vandermonde −1 , for generic matrices, no closed-form formulas are known for the entries of Kx,y k. Indeed, the inversion formulas presented in [9] involve a polynomial division, and the inversion algorithms introduced in [12, 18, 19] have a recursive character, −1 namely, the entries of Kx,y are computed according to suitable orderings. Hence, it is not obvious how to study the structured conditioning of Kx,y by a straightforward use of the deﬁnitions. In the sequel we will exploit a factorization approach analogous to the one already introduced in the preceding section. Lemma 3. Let ξ = (ξ1 , . . . , ξn−k )T , ξj = ei(j−1)2π/(n−k) . Furthermore, let a = (xn−k − 1, . . . , xn−k − 1)T and yˆ ∈ Cn , yˆ = (y1 , . . . , yk , ξ1 , . . . , ξn−k )T . Let n 1

I O T = , (16) (n − k)−1/2 Cξ,y Φ−1

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partitioned as N0 , i.e., the Cauchy matrix Cξ,y has order (n − k) × k and Φ = (n − k)−1/2 Vξ is the Fourier matrix of order n − k. Furthermore, let f be the solution of the linear system T T f = en . If Δ(x, yˆ) < +∞ then Kx,y = Da Cx,ˆy Df T. Proof. It is a simple task to verify that I T −1 = −(n − k)−1/2 ΦCξ,y

O Φ

.

(17)

Moreover, since Φ−1 e1 = (n − k)−1/2 1 and Φ−1 P0 Φ = Dξ , we have

I O Dy O √ T N0 T −1 = e1 1T P0 n − kΦ−1 (n − k)−1/2 Cξ,y

I O × −(n − k)−1/2 ΦCξ,y Φ

Dy O = (n − k)−1/2 [Cξ,y Dy − Dξ Cξ,y + 11T ] Φ−1 P0 Φ = Dyˆ. Hence T −1 DyˆT = N0 . Since diagonal matrices commute, we obtain: LDx ,Dyˆ (Da−1 Kx,y T −1 )

= Da−1 [Dx Kx,y − Kx,y N0 ]T −1 = Da−1 aeTn T −1 = 11T Df .

The hypothesis stated on x implies both the invertibility of Da and that of the operator LDx ,Dyˆ . We obtain Da−1 Kx,y T −1 = Cx,ˆy Df , whence thesis follows. We remark that det(Kx,y ) = 0 if all entries from x and y are distinct, see, e.g., [9, Thm. 3.1] or [19]. As a consequence, by Binet-Cauchy Theorem we see that, in the hypotheses of the preceding lemma, we have det(Df ) = 0, that is, f has no zero entries. We consider in the following theorem the structured conditioning of the −1 columns of Kx,y with respect to perturbations in the vector x. We will use some further notations: For any x = (x1 , . . . , xn )T ∈ Cn , consider the two subvectors x(1) = (x1 , . . . , xk )T ∈ Ck ,

x(2) = (xk+1 , . . . , xn )T ∈ Cn−k ,

where k is the integer appearing in the deﬁnition of the Cauchy-Vandermonde matrix under consideration. Furthermore, introduce the relative distance ˆ x˜) = max{δ(x(1) , x δ(x, ˜(1) ), x(2) − x ˜(2) /x}, and consider the following condition measure, naturally induced by it: ˆ (x), F (˜ δ(F x)) . m(F, ) x) = lim sup →0 δ(x,˜ δ(x, x ˜ ) x)≤

(18)

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217

ˆ x Remark that x − x ˜/x ≤ δ(x, ˜) ≤ δ(x, x ˜), and m(F, x) ≤ m(F, ) x).

(19)

Theorem 4. Let 1 ≤ i ≤ n be ﬁxed, and let Fi : Cn → Cn be the function deﬁned −1 as Fi (x) = Kx,y ei . Moreover, let 0 ≤ θ < 2π be arbitrary and let ω = e−iθ . Then,

√ √ γ √ m(F ) i , x) ≤ + n − k (γ + n − k) n−k 1 1 1 (n − k)xn−k 1 1 1 i × Δ(n, ωx, ξ) + 1 n−k 1 , i(n−k)θ 1 1 xi −e where γ = Cξ,ωy , ξ = (ξ1 , . . . , ξn−k ), ξj = ei(j−1)2π/(n−k) and Δ(n, ωx, ξ) can be derived from (11). Proof. As in Theorem 2, we can reduce the analysis to the case θ = 0 by simply considering the rotated vectors x ˆ = ωx and yˆ = ωy, by virtue of (15). In the notations of Lemma 3 we have −1 −1 −1 Kx,y = T −1Df−1 Cx,ˆ y Da .

Recall that f has no zero entries, hence Df is invertible. We obtain Fi (x) =

1 xn−k i

−1

−1 T −1 Df−1 Cx,ˆ y ei .

Then, Fi (x) = G(Hi (x)), where G(x) = T −1x and −1 Hi (x) = (xn−k − 1)−1 Df−1 Cx,ξ ei . i

Using (18), it is straightforward to check that m(F ) i , x) ≤ m(G, ) Hi (x))c(Hi , x). Let u = G(x) and u ˜ = G(˜ x), for arbitrary x, x ˜. In order to bound the quantity ˆ u δ(u, ˜), observe ﬁrstly that δ(u(1) , u ˜(1) ) = δ(x(1) , x ˜(1) ) ≤ δ(x, x ˜). Using u ≥ x/T , we have: (n − k)−1/2 ΦCξ,y (x(1) − x ˜(1) ) + Φ(x(2) − x ˜(2) ) x

(1) √ x(1) − x x(2) − x ˜ ˜(2) + n−k ≤ T γ x x √ ≤ T (γ + n − k)δ(x, x˜). √ √ Since T ≤ γ/ n − k + n − k we have √ √ √ ˆ u ˜). δ(u, ˜) ≤ (γ/ n − k + n − k)(γ + n − k)δ(x, x u(2) − u ˜(2) u

≤

From (18) we obtain

T

√ √ √ m(G, ) x) ≤ (γ/ n − k + n − k)(γ + n − k).

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Furthermore, if we let δ(x, x ˜) = , we have from Theorem 1 x)| ≤ |Hi (x) − Hi (˜

≤

−1 |xn−k − 1|−1 |Df−1 ||Cx,ξ ei − Cx˜−1 i ,ξ ei | 1 1 1 1 1 1 1 1 −1 −1 + 1 n−k − n−k 1 |Df Cx˜,ξ ei | 1 xi −1 x ˜i − 11 −1 |xn−k − 1|−1 |Df−1 Cx,ξ ei |Δ(n, x, ξ) i

| −1 −1 (n − k)|xn−k i |Df Cx,ξ ei | + o() n−k 2 |xi − 1| 1 1 1 (n − k)xn−k 1 1 1 Δ(n, x, ξ)|Hi (x)| + 1 n−k i 1 |Hi (x)| + o(), 1 xi −1 1 +

≤ whence we obtain

1 1 1 (n − k)xn−k 1 1 1 c(Hi , x) ≤ Δ(n, x, ξ) + 1 n−k i 1 , 1 xi −1 1

and the proof is complete.

Remark that an upper bound for the constant γ appearing in the preceding theorem is obtained as follows: k 1 k γ = max ≤ = kΔ(ξ, ωy). 1≤i≤n−k |ξi − ωyj | mini,j |ξi − ωyj | j=1 We omit the proof of the following corollary, as it is essentially the same as that of Corollary 3, in the light of (19): −1 Dd ), where Dd is Corollary 6. Let F : Cn → Cn be deﬁned as F (x) = Vec(Kx,y an arbitrary diagonal matrix whose entries do not depend on x. Furthermore, let ξ = (ξ1 , . . . , ξn−k ), ξj = ei(j−1)2π/(n−k) . For any 0 ≤ θ < 2π, let ω = e−iθ . Then, √ √ √ m(F, x) ≤ (γ/ n − k + n − k)(γ + n − k) 1 1 1 1 xn−k 1 1 i × Δ(n, ωx, ξ) + (n − k) max 1 n−k 1 , i(n−k)θ 1 i 1x − e i 2

where γ = Cξ,ωy and Δ(n, ωx, ξ) can be obtained from (11).

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[21] S. Serra Capizzano; An elementary proof of the exponential conditioning of real Vandermonde matrices. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 10 (2007), 761–768. [22] J.-G. Sun; Bounds for the structured backward errors of Vandermonde systems. SIAM J. Matrix Anal. Appl. 20 (1999), 45–59. [23] E.E. Tyrtyshnikov; Singular values of Cauchy-Toeplitz matrices. Linear Algebra Appl. 161 (1992), 99–116. [24] E.E. Tyrtyshnikov; How bad are Hankel matrices? Numer. Math. 67 (1994), 261–269. [25] H. Xiang and Y. Wei; Structured mixed and componentwise condition numbers of some structured matrices. J. Comput. Appl. Math. 202 (2007), 217–229. Enrico Bozzo and Dario Fasino Dipartimento di Matematica e Informatica Universit` a di Udine Via delle Scienze, 208 I-33100 Udine, Italy e-mail: [email protected] [email protected]

Operator Theory: Advances and Applications, Vol. 199, 221–227 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Factorizations of Totally Negative Matrices V. Cort´es and J.M. Pe˜ na In memory of Georg Heinig

Abstract. A matrix is called totally negative if all its minors are negative. In this paper we characterize two decompositions of totally negative matrices: the QR decomposition and the symmetric-triangular decomposition. Mathematics Subject Classiﬁcation (2000). 15A23; 65F25; 15A48; 15A15. Keywords. Totally negative matrix, QR decomposition, symmetric-triangular decomposition.

1. Introduction and background This paper deals with two factorizations of totally negative matrices relevant in numerical analysis: QR decomposition and symmetric-triangular decomposition (see [9]). A matrix is totally negative (TN) if all its minors are negative. Totally negative matrices belong to the class of N -matrices (matrices with all principal minors negative), which play an important role in Economy (see [3], [10] and [13]). Others aspects of totally negative matrices have been considered in [2], [5] and [7]. Besides, they belong to the class of strictly sign regular matrices, which will be deﬁned below and which are very important in many applications due to their property known as variation diminishing property (see [1], [11] and [12]). Let us start with basic notations. Following [1] and [6], for k, n ∈ N , 1 ≤ k ≤ n, Qk,n will denote the set of all increasing sequences of k natural numbers not greater than n. For each k−1 α ∈ Qk,n , its dispersion number d(α) is deﬁned by d(α) := i=1 (αi+1 − αi − 1) = αk − α1 − (k − 1), with the convention d(α) = 0 for α ∈ Q1,n . Let us observe that d(α) = 0 means that α consists of k consecutive integers. For α = (α1 , α2 , . . . , αk ), β = (β1 , β2 , . . . , βk ) ∈ Qk,n and A an n × n matrix, we denote by A[α|β] the k × k submatrix of A containing rows α1 , α2 , . . . , αk and columns β1 , β2 , . . . , βk of A. If α = β, we denote by A[α] := A[α|α] the corresponding principal minor. A column-initial minor of A is a minor of the form

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det A[α|1, . . . , k], where α ∈ Qk,n with d(α) = 0 and 1 ≤ k ≤ n. Analogously, a row-initial minor of A is a minor of the form det A[1, . . . , k|α], where α and k are given as above. Let A be an n × n lower (resp., upper) triangular matrix. Following [4], the minors det A[α|β] with αk ≥ βk (resp., αk ≤ βk ∀k) are called nontrivial minors of A because all the remaining minors are obviously equal to zero. A triangular matrix A is called ΔTP if its nontrivial minors are all positive. We remark that these matrices are also called ΔSTP in other papers (see [8]). By a signature sequence we mean an (inﬁnite) real sequence ε = (εi ) with |εi | = 1, i = 1, 2, . . . . An n × n matrix A verifying εk det A[α|β] > 0 for all α, β ∈ Qk,n and k = 1, . . . , n is called strictly sign regular with signature ε1 , . . . , εn , and will be denoted by SSR. Two important subclasses of the strictly sign regular matrices are the totally positive matrices (with all its minors positive) and the totally negative matrices (deﬁned above). This terminology is more frequent nowadays, although these matrices also have been called in the literature as strictly totally positive and strictly totally negative, respectively. The ﬁrst part of the following result comes from Theorem 3.1 of [4] and the second part is provided by Remark 3.6 of [7]. Theorem 1.1. (i) A lower (resp., upper) triangular matrix M is ΔTP if and only if all columninitial (resp., row-initial) minors of M are positive. (ii) Let A = (aij )1≤i,j≤n be a nonsingular matrix and ann < 0. Then A is TN if and only if all its initial minors are negative. An LDU -factorization of a matrix A is the decomposition A = LDU where L (resp., U ) is a lower (resp., upper) triangular, unit diagonal matrix (i.e., with all diagonal entries equal to 1), and D is a diagonal matrix. From now on, A = LDU will refer to this decomposition. Let us recall the following result, which will be used in the following sections. The proof can be seen in Proposition 2.1 of [7]. Theorem 1.2. If A is SSR, then A = LDU with L (resp., U ) a ΔTP and lower (resp., upper) triangular, unit diagonal matrix and D a diagonal nonsingular matrix. Finally, let us recall the well-known Cauchy-Binet formula. If A, B are n × n matrices, then we have the following determinantal identity: det(AB)[α|β] = det A[α|w] det B[w|β], α, β ∈ Qk,n . w∈Qk,n

Section 2 contains some auxiliary results. In Section 3 we characterize the QR factorization of totally negative matrices. Section 4 contains the characterization of the symmetric-triangular factorization of totally negative matrices and includes an example showing that a condition used in our characterizations (the fact that ann < 0) cannot be suppressed.

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2. Auxiliary results We are going to deﬁne some special classes of matrices which will play a key role in the characterizations of this paper. Deﬁnition 2.1. A nonsingular matrix A is said to be lowerly TN if it can be decomposed in the form A = LDU and LDΣ is ΔTP, where Σ is a diagonal matrix with diagonal entries −1, +1, . . . , +1. If, in addition, U −1 satisﬁes that ΣU −1 Σ is ΔTP, then the matrix is called strict γ − -matrix. In the previous deﬁnitions, since L is unit diagonal, the fact that LDΣ is ΔTP implies that L and DΣ are ΔTP. The following result relates the two concepts introduced in the previous definition. Proposition 2.2. If A = LDU and (AT )−1 are lowerly TN, then A is a strict γ − matrix. Therefore, an orthogonal matrix is lowerly TN if and only if it is strict γ − -matrix. Proof. We have to see that, under the conditions of the proposition, ΣU −1 Σ is ˜D ˜U ˜ we get ΔTP. From the factorizations A = LDU and (AT )−1 = L T −1 T T −1 T ˜ ˜ ˜ (U ) = DL (A ) = DL (LDU ). (2.1) Since A and (AT )−1 are lowerly TN, we have from (2.1) that ˜ DΣ)(Σ ˜ ˜ Σ), U Σ(U T )−1 Σ = (ΣDLT )(L

(2.2) ˜ ˜ where the upper triangular matrix ΣDL and the lower triangular matrix LDΣ ˜ are ΔTP. Moreover, ΣU Σ is an upper triangular, unit diagonal matrix. By the Cauchy-Binet formula, for all α ∈ Qk,n with 1 ≤ k ≤ n and d(α) = 0 we can derive from (2.2) T

det B[α|1, . . . , k] = det C[α|1, . . . , k] det F [1, . . . , k] = det C[α|1, . . . , k], (2.3) ˜ DΣ) ˜ ˜ Σ. where B := Σ(U T )−1 Σ, C := (ΣDLT )(L and F := ΣU Again by the Cauchy-Binet formula, for all α ∈ Qk,n with d(α) = 0, we have ˜ DΣ)[β|1, ˜ det(ΣDLT )[α|β] det(L . . . , k]. (2.4) det C[α|1, . . . , k] = β∈Qk,n

˜ DΣ ˜ are ΔTP, we obFrom (2.4) and taking into account that ΣDLT and L tain det C[α|1, . . . , k] > 0 for all α ∈ Qk,n with d(α) = 0. Then, from (2.3) det B[α|1, . . . , k] > 0 for all α ∈ Qk,n with d(α) = 0. Therefore, all column-initial minors of the lower triangular matrix B are positive and so, by Theorem 1.1 (i), B = Σ(U T )−1 Σ is ΔTP, which implies ΣU −1 Σ is ΔTP. The following result for lowerly TN matrices will be very useful in this paper. Proposition 2.3. Let A be an n × n matrix. If A = CV with C lowerly TN and V upper triangular with positive diagonal, then all column-initial minors of A are negative.

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Proof. Since C is lowerly TN, we have that C = LC DC UC and the matrix LC DC Σ is ΔTP, where Σ is a diagonal matrix with diagonal entries −1, +1, . . . , +1. Then by the Cauchy-Binet formula, we get for k = 1, . . . , n 0 < det(LC DC Σ)[α|1, . . . , k] = det(LC DC )[α|1, . . . , k] det Σ[1, . . . , k] = − det(LC DC )[α|1, . . . , k] for all α ∈ Qk,n and d(α) = 0. So, the sign of det(LC DC )[α|1, . . . , k] is −1. Again by the Cauchy-Binet formula, we get det C[α|1, . . . , k] = det(LC DC )[α|1, . . . , k] det UC [1, . . . , k] = det(LC DC )[α|1, . . . , k] for all α ∈ Qk,n and d(α) = 0. So, all column-initial minors of order k of C are negative. Applying again the Cauchy-Binet formula to A = CV , we obtain det A[α|1, . . . , k] = det C[α|1, . . . , k] det V [1, . . . , k]

(2.5)

for all α ∈ Qk,n and d(α) = 0. Since V has positive diagonal entries, we deduce from (2.5) that all column-initial minors of A are negative.

3. QR decomposition of TN matrices The following theorem gives a characterization of TN matrices by their QR factorization. Theorem 3.1. Let A be an n × n matrix. A is TN if and only if ann < 0 and ˜ R, ˜ (3.1) A = QR, AT = Q − ˜ ˜ where Q and Q are orthogonal and strict γ -matrices and R and R are upper triangular and ΔTP matrices. Proof. If B is a TN matrix, then bnn < 0. Since B is nonsingular, B = QR with Q orthogonal and R upper triangular with positive diagonal entries. If LQ DQ UQ is the LDU -factorization of Q, then we have B = LQ DQ (UQ R).

(3.2) ¯U ¯, Since R is upper triangular with positive diagonal, we can write UQ R = D ¯ ¯ where U is upper triangular, unit diagonal matrix and D is a diagonal matrix with positive diagonal. So, from (3.2) we get ¯ U ¯. (3.3) B = LQ (DQ D) Since B is TN (so that, B is SSR with signature εi = −1 for all i, by Theorem 1.2, B = LDU with D a diagonal nonsingular matrix and L (resp., U ) ΔTP and lower (resp., upper) triangular with unit diagonal. The uniqueness of the LDU ¯ U =U ¯ , and so LQ is ΔTP. factorization implies by (3.3) that LQ = L, D = DQ D, The diagonal entries di of D satisfy det B[1, . . . , i] di = (3.4) det B[1, . . . , i − 1]

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and so sign(di ) = εi /εi−1 = +1 for i = 2, . . . , n, and sign(d1 ) = ε1 /ε0 = −1, taking ε0 := +1. Hence, if Σ is a diagonal matrix with diagonal entries −1, +1, . . . , +1, ¯ has positive diagonal entries and so DQ Σ is also a diagonal matrix DΣ = DQ DΣ with positive diagonal. Therefore, Q is lowerly TN and by Proposition 2.2 it is a strict γ − -matrix. Since B = QR, (3.5) B T B = RT R and RT R can be computed from R by multiplying each row by a positive number and adding a linear combination of the previous rows. Therefore, the row-initial minors of RT R have the same sign as the corresponding minors of R. Since the product of two TN matrices is TP by Theorem 3.1 of [1], we deduce from (3.5) that RT R is TP. Then the row-initial minors of RT R and R are positive, and by Theorem 1.1 (i), R is ΔTP. Now, since A and AT are TN, applying the previous reasoning to A and AT , (3.1) follows with the stated properties. Conversely, applying Proposition 2.3 to the matrices A and AT with C = ˜ ˜ (respectively), we conclude that all the initial minors of A Q, Q and V = R, R are negative. Then, taking into account that ann < 0, by Proposition 1.1 (ii) A is TN.

4. Symmetric-triangular decomposition of TN matrices Recently, Golub and Yuan have introduced in [9] a symmetric-triangular decomposition of a nonsingular matrix. We now characterize TN matrices in terms of this decomposition. Theorem 4.1. Let A be an n × n matrix. A is TN if and only if ann < 0 and ˜ A = SR, AT = S˜R, (4.1) ˜ are upper where S and S˜ are symmetric and lowerly TN matrices and R and R triangular matrices with unit diagonal. Proof. Let B be a TN matrix. By Theorem 1.2, B = LDU with D a diagonal nonsingular matrix and L (resp., U ) ΔTP and lower (upper) triangular with unit diagonal. If we write B = (LDLT )(LT )−1 U , then we can consider S := LDLT , R := (LT )−1 U,

(4.2)

where S is a nonsingular and symmetric matrix and R is an upper triangular matrix with unit diagonal. Now, let us see S is lowerly TN. So, we have to prove that LDΣ is ΔTP, where Σ is a diagonal matrix with diagonal entries −1, +1, . . . , +1. Reasoning through (3.4) as in the proof of Theorem 3.1, it can be deduced that DΣ is a diagonal matrix with positive diagonal. So, LDΣ is ΔTP because L is ΔTP. Then, from (4.2), we conclude that S is lowerly TN.

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Since the matrices A and AT are TN, applying the previous reasoning to A and AT , the necessary condition holds. Conversely, if we apply Proposition 2.3 to the matrices A and AT with C = ˜ ˜ (respectively), then we conclude that all the initial minors of S, S and V = R, R A are negative. Then, taking into account that ann < 0, by Proposition 1.1 (ii) A is TN. The following example shows that the condition ann < 0 cannot be suppressed in theorems 3.1 and 4.1. Example. Let A be a nonsingular matrix given by

−1 −1 . A= −1 0 Then A = QR with Q orthogonal and R upper triangular with positive diagonal entries given by √ √ √ √ −1/√2 −1/√ 2 2 1/√2 Q= , R= . −1/ 2 1/ 2 0 1/ 2 We also have AT = A = QR. The matrix R is ΔTP. Besides, Q = LQ DQ UQ where √

1 0 −1/ 2 √0 1 1 , DQ = . , UQ = LQ = 1 1 0 1 2 0 Clearly the matrices

√ 1/√2 L Q DQ Σ = 1/ 2

√0 , 2

ΣU

−1

1 1 , Σ= 0 1

with Σ = diag{−1, +1}, are ΔTP, and so Q is strict γ − -matrix. However, A is not TN because det A[2] = 0. As for the symmetric-triangular decomposition, we have A = SR with S symmetric and R upper triangular with unit diagonal given by

1 0 −1 −1 . , R= S= 0 1 −1 0 We also have AT = A = SR. The 1 L= 1

matrix S = LDLT where

−1 0 0 . , D= 0 1 1

1 LDΣ = 1 with Σ = diag{−1, +1}, is ΔTP, S is lowerly

Since the matrix

0 1 TN. However, A is not TN.

Acknowledgment This work is partially supported by the Spanish Research Grant MTM2009-07315 and by Gobierno de Arag´ on and Fondo Social Europeo.

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References [1] T. Ando, Totally positive matrices, Linear Algebra Appl. 90 (1987) 165–219. [2] C.M. Ara´ ujo, J.R. Torregrosa, A.M. Urbano, Totally nonpositive completions on partial matrices, Linear Algebra Appl. 413 (2006), 403–424. [3] R.B. Bapat, T.E.S. Raghavan, Nonnegative Matrices and Applications, Cambridge University Press, New York, 1997. [4] C.W. Cryer, LU-factorization of totally positive matrices, Linear Algebra Appl. 7 (1973) 83–92. [5] S.M. Fallat, P. Van Den Driessche, On matrices with all minors negative, Electron. J. Linear Algebra 7 (2000), 92–99. [6] J. Garloﬀ, Intervals of almost totally positive matrices, Linear Algebra Appl. 363 (2003) 103–108. [7] M. Gasca and J.M. Pe˜ na, A test for strict sign-regularity, Linear Algebra Appl. 197 (1994) 133–142. [8] M. Gasca and J.M. Pe˜ na, A matricial description of Neville elimination with application to total positivity, Linear Algebra Appl. 202 (1994) 33–54. [9] G.H. Golub and J.Y. Yuan, Symmetric-triangular decomposition and its applications. I. Theorems and algorithms, BIT 42 (2002) 814–822. [10] T. Parthasarathy and G. Ravindran, N -matrices, Linear Algebra Appl. 139 (1990), 89–102. [11] J. M. Pe˜ na (Ed.), Shape preserving representations in Computer-Aided Geometric Design, Nova Science Publishers, Commack (New York), 1999. [12] J.M. Pe˜ na, On nonsingular sign regular matrices, Linear Algebra Appl. 359 (2003), 91–100. [13] R. Saigal, On the class of complementary cones and Lemke’s algorithm, SIAM J. Appl. Math. 23 (1972), 46–60. V. Cort´es and J.M. Pe˜ na Depto. Matem´ atica Aplicada Universidad de Zaragoza E-50009 Zaragoza, Spain e-mail: [email protected] [email protected]

Operator Theory: Advances and Applications, Vol. 199, 229–254 c 2010 Birkh¨ auser Verlag Basel/Switzerland

QR-factorization of Displacement Structured Matrices Using a Rank Structured Matrix Approach Steven Delvaux, Luca Gemignani and Marc Van Barel Abstract. A general scheme is proposed for computing the QR-factorization of certain displacement structured matrices, including Cauchy-like, Vandermonde-like, Toeplitz-like and Hankel-like matrices, hereby extending some earlier work for the QR-factorization of the Cauchy matrix. The algorithm employs a chasing scheme for the recursive construction of a diagonal plus semiseparable matrix of semiseparability rank r, where r is equal to the given displacement rank. The complexity is O(r2 n2 ) operations in the general case, and O(rn2 ) operations in the Toeplitz- and Hankel-like case, where n denotes the matrix size. Numerical experiments are provided. Mathematics Subject Classiﬁcation (2000). 65F18, 15A23, 15A03. Keywords. Displacement structures, QR-factorization, lower semiseparable plus diagonal matrices, chasing procedure.

1. Introduction In this paper we present a novel uniﬁed approach for computing the QR-factorization of displacement structured matrices, including some cases of main interest, namely Vandermonde-like, Toeplitz-like and Hankel-like matrices. Cauchy-like maThe research of the ﬁrst and third author was partially supported by the Research Council K.U. Leuven, project OT/05/40 (Large rank structured matrix computations), Center of Excellence: Optimization in Engineering, by the Fund for Scientiﬁc Research – Flanders (Belgium), G.0455.0 (RHPH: Riemann-Hilbert problems, random matrices and Pad´e-Hermite approximation), G.0423.05 (RAM: Rational modelling: optimal conditioning and stable algorithms), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Oﬃce for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identiﬁcation & Modelling). The scientiﬁc responsibility rests with the authors. The ﬁrst author is a Postdoctoral Fellow of the Fund for Scientiﬁc Research – Flanders (Belgium). The work of the second author has been supported by MIUR under project 2004015437.

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trices can also be treated in this framework, provided they have at least one set of data points lying on the unit circle of the complex plane C. Generalizations of the algorithm to deal with Toeplitz-plus-Hankel-like matrices and Cauchy-like matrices with data points on the real axis R are also possible and will be discussed elsewhere. In the literature, some algorithms for the QR-factorization of Toeplitz-like matrices are already known, see [1, 12] and the references therein. These methods typically compute the upper triangular factor by factoring the normal equation matrix via the generalized Schur algorithm. If the input matrix is ill-conditioned or exactly rank deﬁcient, then the computed factorization of these algorithms can be very poorly accurate or even not an exact QR-factorization [12]. In contrast to these existing methods, the algorithms of the present paper proceed by translating the QR-factorization problem for a given displacement structured matrix into an equivalent chasing problem for an associated unitary rank structured matrix. The approach followed in the present paper is motivated by the result in [7], where the QR-factorization of the classical Cauchy matrix with real data points was translated into a chasing problem for an associated Hermitian diagonal plus semiseparable matrix of semiseparability rank one. However, diﬀerently from [7] here we will focus on those displacement structures leading to unitary rank structured matrices, since this is the case for some of the more important classes such as Vandermonde-like, Toeplitz-like matrices and so on. Incidentally, let us point out that also transforms of a Toeplitz matrix into a Cauchy-like matrix with data points lying on the real line R are possible [9, 10, 11], but these will not be covered in the present paper. The unitary rank structured matrices in this paper will be represented and manipulated as a product of Givens transformations [5, 8]. This representation leads to an asymptotically optimal number of O(rn) parameters. Moreover, in contrast to the so-called sequentially semiseparable, quasiseparable, uv and Givensweight representations used to represent rank structured matrices in the literature, the Givens product representation has the advantage that the unitarity of the matrix is an explicit part of the representation. Chasing problems for tridiagonal and banded matrices are generally solved by using bulge-chasing techniques. The rationale for employing these techniques is that band matrices have a speciﬁed sparsity pattern and we can annihilate a bulge by moving it along this pattern downwards or upwards until it disappears by using a sequence of Givens rotations applied as a similarity transform. Rank structured matrices are generally dense so that the occurrence of a bulge in the rank structure is not visible. Our major contribution is to show that appropriate generalizations of bulge-chasing techniques still work for unitary rank structures represented in factored form thus leading to an eﬃcient solution of the associated chasing problem. The cumulative similarity transformation returned as output by the chasing procedure allows to determine the Q-factor of the QR-factorization of the given displacement structured matrix.

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The total complexity of this algorithm for QR-factorization is O(r2 n2 ) operations in the general case, and O(rn2 ) operations in the Toeplitz-like case, where r denotes the displacement rank, and where n denotes the matrix size. It is worth pointing out that the algorithm computes the QR-factorization of the given displacement structured matrix A = QR in full form, i.e., the R-factor is computed as a full upper triangular matrix having (n+1)n entries, while the Q-factor is computed 2 n(n−1) Givens transformations. Hence, the proposed to consist of a full product of 2 scheme does not lead to a sparsity in the representation of the QR-factorization, but rather to an eﬃcient way for computing these parameters. The methods introduced in this paper are also strongly related to the chasing methods for diagonal plus semiseparable matrices in solving inverse eigenvalue problems [15]. More precisely, • we present methods which can be literally used for solving some of the inverse eigenvalue problems too, • we explain the mechanism behind a source of numerical instability, which may illuminate the numerical experiments for inverse eigenvalue problems as well. The remainder of this paper is organized as follows. In Section 2 we establish basic notation and brieﬂy recall some rank structure concepts. In Section 3 we describe the general scheme for the QR-factorization of a displacement structured matrix and its relation to a chasing problem for an associated lower semiseparable plus diagonal matrix. Sections 4, 5 and 6 specify this scheme to the case of Cauchylike, Vandermonde-like and Toeplitz-like matrices. Finally, Section 7 reports on the results of some numerical experiments, and it discusses some interpretations of the observed numerical stability issues.

2. Rank Structure Preliminaries Let us ﬁrst deﬁne in more detail the rank structures arising from the methods of this paper. Deﬁnition 1. We deﬁne a rank structure R on Cn×n as a collection of so-called structure blocks R = {Bk }k . Each structure block Bk is characterized as a 4-tuple Bk = (ik , jk , rk , λk ), where ik is the row index, jk the column index, rk the rank upper bound and λk ∈ C is called the shift element. We say a matrix A ∈ Cn×n to satisfy the rank structure R if for each k, RankAk (ik : n, 1 : jk ) ≤ rk ,

where Ak = A − λk I.

Thus after subtracting the shift element λk from the diagonal entries, we must get a low rank block.

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As a special case, when a structure block Bk has shift element equal to zero, or when it is situated strictly below the main diagonal, then we call it pure. We sometimes denote such a structure block by Bpure,k . Figure 1 shows an example with two structure blocks.

Rk 3 λ1 =0.89

Rk 1

Figure 1. Example of a rank structure with two structure blocks. The left structure block B1 intersects the diagonal and has shift element λ1 = 0.89, while the second structure block B2 is ‘pure’. The notation ‘Rk r’ denotes that the structure block is of rank at most r. As a special case, a matrix is called lower semiseparable plus diagonal of semiseparability rank r if it satisﬁes a set of structure blocks Bk : (ik , jk , rk , λk ) = (k, k, r, λk ), k = 1, . . . , n. Here the λk ∈ C are an integral part of the rank structure. We will sometimes refer to them as the diagonal elements of the rank structure. Another remarkable special structure is deﬁned by Bpure,k : (ik , jk , rk , λk ) = (s + k, k, 0, 0), k = 1, . . . , n − s. If A ∈ Cn×n satisﬁes all the Bpure,k then A has lower bandwidth s−1. Therefore, rank structures are a generalization of band structures. A unitary rank structured matrix admits a condensed representation as the product of Givens rotations. For computational purposes it is useful to describe pictorially this product by means of an associated Givens product representation graph [5]. The following example clariﬁes these concepts. Let A ∈ C6×6 be a unitary lower semiseparable matrix of semiseparability rank 1, i.e., A satisﬁes a set of structure blocks Bk : (ik , jk , rk , λk ) = (k, k, 1, 0), k = 1, . . . , 6. The QRdecomposition of A yields a condensed parametrization of the matrix as a product of Givens transformations A = G5,6 G4,5 · · · G1,2 ; here the notation Gi,i+1 denotes a Givens transformation, i.e., a unitary transformation which equals the identity matrix except for its entries in rows and columns i, i + 1. We represent the above matrix product by the following graph

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Here each little line segment represents a Givens transformation Gi,i+1 ‘acting’ on the rows of an (invisible) matrix standing on the right of it. The row index i of the Givens transformation Gi,i+1 can be derived from the height at which the Givens transformation is standing in the ﬁgure, e.g., the top rightmost line segment in the above ﬁgure corresponds to G1,2 , and so on. Now let us suppose that A is modiﬁed both in the third and in the fourth is still unitary and, moreover, column in such a way that the perturbed matrix A H H is revealed G1,2 · · · G5,6 A = I2 ⊕ G3,4 ⊕ I2 . The ‘bulge ‘in the rank structure of A by the associated graph

At the core of our proposed chasing methods for unitary rank structured matrices is the basic observation that, similarly as in the banded case, this bulge can be moved along the graph by a sequence of unitary similarity transformations determined by swapping techniques applied to short sequences of consecutive Givens rotations.

3. QR-factorization of displacement structured matrices In this section we start with some general theory for the QR-factorization of displacement structured matrices. We will assume that A ∈ Cn×n is an invertible displacement structured matrix, i.e., a matrix satisfying a displacement equation of the form Y A − AZ = Rk r, (1) where Y, Z ∈ Cn×n are ﬁxed coeﬃcient matrices, and where the right-hand side Rk r has to satisfy the constraint that it is of rank at most r, where r ∈ N is the displacement rank. Examples of such displacement equations could be those deﬁning Cauchy-like, Vandermonde-like, Toeplitz-like matrices and so on, each of them having a speciﬁc choice for the coeﬃcient matrices Y and Z as well as for the displacement rank r. The treatment of these speciﬁc classes is deferred to the next sections. Remark 2. Some comments are in order concerning the assumption that A is invertible. This condition seems to preclude our method to be applied in many interesting cases where the matrix is rank-deﬁcient or it is ill conditioned. However, in [2] it has been proved that the assumption can be removed by using a continuity argument and, indeed, our approach still works in the singular case. More precisely, since the QR decomposition of a singular matrix is not essentially unique, we can show that a QR decomposition of A exists such that the Q-factor is the solution of a chasing problem for an associated unitary rank structured matrix.

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Let us now assume a general displacement equation (1). Inserting the QRfactorization A = QR into this equation leads to Y (QR) − (QR)Z = Rk r 6r ⇔ (QH Y Q)R − RZ = Rk 6 6 r, ⇔ (QH Y Q) = RZR−1 + Rk

(2) (3) (4)

6 6 r and Rk 6 r are new matrices of rank at most r. Note that we assumed where Rk −1 here R to exist. Now we would like to use the above relations to obtain an alternative way of computing the QR-factorization A = QR. We will have to assume to this end that • both Y and Z are upper triangular matrices.

(5)

By the above assumption, the matrix RZR−1 occurring in (4) must also be an upper triangular matrix. A closer look reveals that it has diagonal elements precisely equal to those of Z: zi,i , i = 1, . . . , n. Hence, the right-hand side of (4) will be a lower semiseparable plus diagonal matrix of semiseparability rank r, with diagonal elements of the structure precisely equal to the zi,i . In order to exploit this observation, let us partition

y1,1 Y1,2 Z1,1 Z1,2 Y = , Z= , (6) 0 Y2,2 0 z2,2 where y1,1 and z2,2 denote the top left and bottom right elements of Y and Z, respectively. We also partition

A1,1 a1,2 , (7) A= A2,1 A2,2 where a1,2 is the top right element of A. From the assumptions on the upper triangularity of Y and Z, it follows that the submatrix A2,1 inherits the displacement structure from A. Indeed, this follows by evaluating the last block row and ﬁrst block column of the displacement equation (1), and using (6), from which Y2,2 A2,1 − A2,1 Z1,1 = (Rk r)2,1 . Thus the matrix A2,1 is displacement structured too. This suggests a recursive procedure: Assume that a QR-factorization has been computed for the displacement structured matrix A2,1 : A2,1 = QR.

(8)

Clearly, given the knowledge of the QR-factorization (8), the full matrix A in (7) can be transformed into a Hessenberg matrix H, and hence its QR-factorization can be completed by applying a single downgoing sequence of Givens transformations to the rows.

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On the other hand, we know that the QR-factorization (8) leads to an associated lower semiseparable plus diagonal equation (4), i.e., QH Y2,2 Q = S,

(9)

S := RZ1,1 R−1 + Rk r

(10)

where is already lower semiseparable plus diagonal of semiseparability rank r, with diagonal elements of the structure equal to zi,i , i = 1, . . . , n − 1. By embedding this relation in a full n by n form, it follows that

y1,1 Y1,2 Q y1,1 Y1,2 1 0 1 0 . (11) = 0 S 0 Y2,2 0 Q 0 QH We should stress that (11) has been obtained from the QR-factorization (8) of size n−1. But we would like now to bring it to the required form (4), induced by the QR-factorization of the full n by n matrix A. It was already mentioned before that this aim can be achieved by incorporating a downgoing sequence of Givens H H transformations GH n−1,n , . . . , G1,2 into the matrix Q . Applying these transformations to both unitary factors in the left-hand side of (11), and applying them also to the right-hand side of (11) transforms the latter to

y1,1 Y1,2 Q H G1,2 · · · Gn−1,n . · · · G (12) S˜ := GH n−1,n 1,2 0 S This relation (12) is the basis of the chasing procedure of this paper. It means that given the lower semiseparable plus diagonal matrix S of size n − 1, we must apply a similarity operation with a downgoing sequence of Givens transformations, such that the resulting matrix S˜ in (12) is lower semiseparable plus diagonal of size n. Under quite mild assumptions on the rank structure of S it can be proved [2] that the unitary Hessenberg matrix G1,2 · · · Gn−1,n satisfying (12) is essentially unique. Roughly speaking, this means that the Q-factor in the QR decomposition (8) of A2,1 can be updated in an essentially unique way to obtain a solution Q of (4). Whence, this latter Q must be the Q-factor of a certain QR factorization of the full matrix A in (7). The above description, and in particular (12), suggests that we could use a structure-preserving chasing procedure to determine the subsequent Givens transformations Gi,i+1 , i = 1, . . . , n − 1. In fact, one could retrieve more details about the intended chasing procedure, by comparing the position of the diagonal elements of the structure zi,i before and after the chasing, cf. (10). This reveals that each of the original diagonal elements of the structure zi,i , i = 1, . . . , n − 1 should be chased one position upwards, while a new diagonal element zn,n is ‘installed’ at the bottom of the matrix (12). To manipulate the lower diagonal plus semiseparable matrix (12) eﬃciently during the chasing scheme, we would like that not only the lower, but also the

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upper triangular part of this matrix has bounded ranks. We will realize this by assuming that • Y is a unitary/Hermitian plus low rank matrix.

(13)

Indeed, it is known that this type of structure is preserved under unitary similarity transformations, and moreover, that the rank structure in the lower triangular part of such matrices, induces rank structure in the upper triangular part as well. Applying this to the lower semiseparable plus diagonal structure of the matrix S in (9) shows that this matrix must be upper semiseparable plus diagonal as well, or semiseparable plus diagonal for short. This will allow the use of eﬃcient representations to represent the matrix S during the chasing scheme, such as sequentially semiseparable, quasiseparable, uv, or Givens-weight representations, see, e.g., [6, 3]. The best choice of representation depends on many factors. For example, for each of the practical examples of the next sections we will have that the matrix Y , and hence the matrix S in (9), are purely unitary matrices, and consequently we will make use there of Givens product representations [5]. Incidentally, note that for the special case where Y in (13) is exactly unitary or Hermitian, combining this with the upper triangularity of Y reveals that Y must be diagonal. However, note that the diagonality of Y does not imply the diagonality of the matrix S in (9), since the latter is semiseparable plus diagonal with semiseparability rank determined by the displacement rank, cf. (10). One element is still missing in the above scheme. Namely, there is the fact that the structure-chasing procedure will unavoidably break down for the determination of the ﬁrst r, as well as for the ﬁnal r Givens transformations Gi,i+1 in (12), since the corresponding Rk r structure blocks are ‘trivially satisﬁed’, and hence preserved by any Givens transformation Gi,i+1 . Hence, we should be able to ﬁnd these Givens transformations in a direct way, using the fact that they belong to the QR-factorization of A. To this end, recall the Hessenberg matrix

a1,2 A1,1 (14) H= R QH A2,2 into which the matrix A has already been brought by means of the unitary matrix QH of size n − 1, where R := QH A2,1 is already upper triangular. We should then be able to derive the ﬁrst and last r columns of this Hessenberg matrix, since the action of making these columns upper triangular will reveal the required Givens transformations Gi,i+1 needed for doing this. Let us assume by induction that we have already knowledge of the ﬁrst and last r columns of the submatrix R in (14). It will clearly suﬃce if we can determine the required elements of the ﬁrst row and last column of the matrix H in (14). Let us begin with the ﬁrst row of H. It is clear from (14) that this row is equal to that of the original displacement structured matrix A. Due to the supposed upper triangularity of Y and Z, we will be able to derive these entries by means of the original displacement equation (1) which the matrix A has to satisfy, which

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we restate here for convenience: Y A − AZ = Rk r.

(15)

Indeed, evaluating the last row and ﬁrst column of (15) reveals that (yn,n − z1,1 )an,1 = (Rk r)n,1 , from which an,1 can be derived. One can then gradually proceed towards the top right direction of the matrix A, using a recursive scheme to determine the subsequent entries of this matrix. Note that we have to assume here that the displacement operator is invertible, i.e., that yi,i = zj,j for all i and j. It is clear that the complexity of determining all entries of the matrix A in this way will require O(n3 ) operations in the general case, which we consider to be unacceptably expensive. In order to obtain a better complexity, we assume from now on that, in addition to being upper triangular, we have that • both Y and Z have upper bandwidth at most b, with b ≤ r. Using this assumption, the complexity of determining all entries of A from its displacement equation is easily seen to reduce to O(b2 n2 ) operations. We remark that we need each time only those entries in the ﬁrst and last r columns of the top row of (14), but in the process of determining the latter, the computation of the elements in the other columns seems to be an unavoidable step. determined one by one in a recursive way. (Exception to this is when the bandwidth b = 0, thus in case where Y and Z are diagonal matrices, in which case each entry of the displacement structured matrix A can be determined independently.) We mention also that the condition that Y and Z have limited bandwidth can be replaced by the more general condition that they are rank structured, but we will not go further into this. Now we consider the computation of the last column of H in (14). To this end, let us characterize this Hessenberg matrix in an alternative way. From the decomposition A = QH (we write here Q instead of 1 ⊕ Q by abuse of notation), one can derive in an analogous way to (2), (3) that ⇔

Y (QH) − (QH)Z = Rk r 6 r, (QH Y Q)H − HZ = Rk

6 r is a new matrix of rank at most r. Evaluating the last column of this where Rk last equation leads to [(QH Y Q) − zn,n I]Hcol

n

=

n−1

zk,n Hcol

k

6 r)col n , + (Rk

(16)

k=n−b

where the subscript col k is used to denote the kth column of a matrix, and where b denotes again the supposed bandwidth of the coeﬃcient matrix Z. This equation can be used to solve for the last column Hcol n in terms of the previous columns of H. Note that the coeﬃcient matrix QH Y Q occurring in (16) is nothing but the lower diagonal plus semiseparable matrix S used during the chasing procedure, cf. (9). Since this matrix is rank structured, it will be possible to solve (16) in an eﬃcient way; see further.

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Summarized, we have now speciﬁed how to determine the entries of the ﬁrst and last r columns of the matrix H in (14), whose information is important for determining the ﬁrst r and last r Givens transformations Gi,i+1 in the chasing scheme of (12). This ends the description of one inductive step n − 1 → n in the QR-factorization of the displacement structured matrix A. At the end of performing all these inductive steps k − 1 → k, k = 2, . . . , n, we will ﬁnally have obtained the Givens transformations constituting the Q-factor of the QR-factorization A = QR. To end the description of the method, we should still specify how to compute the subsequent columns of the R-factor of the QR-factorization. But this can be easily done by solving again the subsequent columns of the relation (3). Note that doing this for all columns of R will lead to the same order of complexity as computing all the column vectors Hcol n , during each of the inductive steps n − 1 → n, as described above. We point out that the complexity of computing all the columns of the R-factor of the QR-factorization, and hence the complexity of computing each of the column vectors Hcol n during the algorithm, typically requires O(r2 n2 ) operations, and O(rn2 ) operations in the Toeplitz-like case. Hence, these computations typically form the dominant term in the total complexity of the QR-factorization, since the chasing steps usually require only O(rn2 ) operations. In the next 3 sections we will show how the general scheme for the QRfactorization of displacement structured matrices described in the current section can be speciﬁed to the displacement structures of most interest.

4. The Cauchy-like case In this section we describe the QR-factorization in case where A ∈ Cn×n is a Cauchy-like matrix: Dy A − ADz = Rk r, (17) where the coeﬃcient matrices Dy , Dz ∈ Cn×n are diagonal matrices whose diagonal entries yi,i , zj,j are called data points, with yi,i = zj,j for all i, j, and where the displacement rank r is supposed to be known; note that r = 1 in case of a

1 classical Cauchy matrix A = . yi,i − zj,j i,j=1,...,n It follows from the general scheme of Section 3 that the matrices S in (10) will now be lower diagonal plus semiseparable matrices, with diagonal elements of the structure given by the zi,i . In fact, for reasons that will become clear soon, we will prefer to keep these diagonal elements out of the representation, and to work instead with the induced pure rank structure of S, which is situated just below the main diagonal: see Figure 2. For the scheme to be described in this section to be of practical interest, we will need the following speciﬁcation of (13): • Dy is unitary.

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Rk 3 Rk 3 Rk 3 Rk 3 Rk 3 Rk 3 Rk 3 Rk 3 Rk 3 Rk 3 Rk 3

Figure 2. Induced pure rank structure for a lower diagonal plus semiseparable matrix S of semiseparability rank r = 3. This unitarity will imply the matrices S in (9) to be unitary as well. Thus, the chasing techniques of this section will be expressed in terms of unitary diagonal plus semiseparable matrices. We turn now to the practical representation and manipulation of the unitary diagonal plus semiseparable matrix S of semiseparability rank r. From Section 2 this can be achieved by means of a suitable Givens product representation graph [5]. In the present case, the representation takes the form of Figure 31 .

Figure 3. Givens product representation graph for a unitary diagonal plus semiseparable matrix S of semiseparability rank r = 3. To explain the speciﬁc pattern formed by the Givens transformations in Figure 3, it is useful to think in terms of the QR-factorization S = QR of the given unitary diagonal plus semiseparable matrix S of semiseparability rank r. Due to the unitarity of S, the R-factor of its QR-factorization can be chosen to be the identity matrix. The above QR-factorization is then equivalent to the equation QH S = I, 1 The speciﬁc pattern formed by the topmost Givens transformations in Figure 3 corresponds to the so-called zero-creating variant of the Givens product representation, as introduced in [5].

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the identity matrix. Thus we should apply to the rows of S a set of unitary operations QH in order to eliminate all entries in the lower triangular part of S. Due to the rank structure of S, this process can be applied eﬃciently via a two step procedure. The ﬁrst step in this process is to peel oﬀ all Givens transformations in the left branch of Figure 3, i.e., to multiply S with the Hermitian transposes of these Givens transformations to the rows, going from bottom to top, and hereby compressing the structure blocks of the given rank structured matrix S into blocks of zeros. Incidentally, this reveals that the width of the left branch of the Givens product representation directly reﬂects the underlying ranks of the rank structure, which is r = 3 in the case of Figure 3. What remains after peeling oﬀ the Givens transformations of the left branch of the representation will be a Hessenberg matrix with three subdiagonals, which is nothing but the right branch of the representation of Figure 3. The fact that the right branch has the same width as the left branch, indicates that the structure is situated here just below the main diagonal. For more details about the representation indicated in Figure 3 we refer to [5]. With the aim of manipulating the Givens product representation, we recall the following result. Lemma 3 (Pull-through lemma). Given a unitary 3 by 3 matrix Q which is factorized as Q = G1,2 G2,3 G1,2 , then there exists a refactorization ˜ 1,2 G ˜ 2,3 . ˜ G Q=G 2,3

See Figure 4.

=

Figure 4. Pull-through lemma applied in the downward direction. One could imagine that the leftmost Givens transformation is really ‘pulled through’ the two rightmost Givens transformations. We will now use the above representation for the unitary diagonal plus semiseparable matrix S to determine the subsequent Givens transformations Gi,i+1 used in the chasing procedure (12). First we will show how to determine the Givens transformations Gi,i+1 with i = r + 1, . . . , n − r − 1; the determination of the ﬁrst and last r Givens transformations Gi,i+1 will be discussed later in this section. From the general scheme of Section 3, we know that the similarity transform with Gi,i+1 , GH i,i+1 has to preserve the diagonal plus semiseparable structure, while each of the diagonal elements of the structure zi,i is chased one position upwards by the chasing procedure.

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We will neglect here the information about the diagonal elements zi,i , and put our focus only on the induced pure structure, situated just below the main diagonal, since the latter structure can be read oﬀ directly from the sparsity pattern of the representation in Figure 3, as explained above. Note that this induced pure structure must be preserved during each step of the algorithm. We will use this latter observation to derive the required Givens transformations. Suppose that we are in the ith step of the chasing process, i = r + 1, . . . , n − r − 1, and that we have to determine a Givens transformation G := Gi,i+1 such that a similarity transformation with G preserves the pure induced rank structure of the unitary diagonal plus semiseparable matrix S. This condition can be ensured by requiring that, after pulling G downwards through the left branch, and GH downwards through the right branch of the representation, by using each time r subsequent applications of the pull-through lemma, then the pulled through Givens transformations must precisely annihilate each other. Stated another way, the pulled through versions of G and GH must be related to each other as H and H H , for a suitable Givens transformation H: see Figure 5.

G

G

HH

H

H

Figure 5. Elementary chasing step for the unitary diagonal plus semiseparable matrix S of semiseparability rank r = 3. The preservation of structure is ensured when the pulled through versions of G and GH are related to each other as H and H H , hence cancelling each other out in the middle of the ﬁgure. We want now to exploit this observation to obtain a practical scheme for determining G. To this end, we will ﬁrst investigate more closely the leftmost of the two pull-through relations between G and H in Figure 5: see Figure 6. By expanding the middle Givens product on the left-hand side of Figure 6 into its full matrix form, it turns out that this relation can be restated in terms of the lower triangularity of a certain 2 by 2 matrix A being preserved by multiplying with G on the left and with H H on the right: See Figure 7. In a similar way, one can consider also the rightmost of the two pull-through relations between G and H in Figure 5. Taking the Hermitian transpose of this relation will lead in exactly the same way to a 2 by 2 lower triangular matrix B,

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S. Delvaux, L. Gemignani and M. Van Barel G = H

H

Figure 6. The ﬁgure expresses the fact that the pulled-through version of G is precisely H, and hence can be removed by multiplying with H H on the right. This guarantees the existence of a refactorization having the shape shown in the right part of the ﬁgure. HH G

A =

Figure 7. Equivalent formulation of Figure 6 in its full matrix form.

whose lower triangularity must be preserved by multiplying with G on the left and with H H on the right. The determination of the required Givens transformations G and H reduces then to the following problem. Problem 4. Given two lower triangular matrices

× × 0 , B= A= × × ×

0 ×

.

We are interested in ﬁnding two non-trivial Givens transformations G and H such that the lower triangularity of A, B is preserved in

× 0 × 0 GAH H = , GBH H = , × × × × where the generically nonzero values are denoted as ×. It is clear that there always exists a trivial solution (G, H) = (I2 , I2 ) to this problem, where I2 is the identity matrix of size 2. But we will be interested in the non-trivial solution. It turns out that this non-trivial solution can be easily obtained. First we are going to determine G. We will search it in the form

c s , G= × ×

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243

where c and s are unknown complex numbers such that |c|2 + |s|2 = 1, and where the values that will be irrelevant in the computation are denoted as ×. We compute

ca1,1 + sa2,1 sa2,2 cb1,1 + sb2,1 sb2,2 GA = , GB = . (18) × × × × Now it is clear that the existence of the column operation H H occurring in Problem 4, is equivalent to the condition 1 1 1 ca1,1 + sa2,1 sa2,2 1 1 1 1 cb1,1 + sb2,1 sb2,2 1 = 0. A trivial case where this equation is satisﬁed is when s = 0. This corresponds to the trivial solution (G, H) = (I2 , I2 ) mentioned before. But recall that we were interested in the non-trivial solution. This non-trivial solution can be obtained by skipping the factor s in the second column of the above determinant, and expanding the remaining part of this determinant as (a1,1 b2,2 − a2,2 b1,1 )c = (a2,2 b2,1 − a2,1 b2,2 )s, or alternatively in matrix-vector form as

0 a1,1 b2,2 − a2,2 b1,1 . = G × a2,2 b2,1 − a2,1 b2,2

(19)

It is clear that, provided at least one of the two components of the vector on the left-hand side of (19) is non-zero, then the required Givens transformation G is essentially uniquely determined from this equation. Given the knowledge of G, one can now determine H as well. Indeed, we know that H should be chosen to restore the lower triangularity of each of the two matrices in (18), i.e., to create a zero in the (1, 2) position of both matrices. We point out that in principle, it does not matter which of these two matrices is taken to determine H, but for stability reasons, one should use the one whose top row has the largest norm, or alternatively a least squares variant containing information of both top rows. We should still clarify one point here. Suppose that we insert the computed value of G in the left part of Figure 6, and subsequently pull it through, so that this Givens transformation is chased to the bottom right. Will it then be guaranteed that we end up with (essentially) the computed value of H at the end of this pullthrough process, so that the factor H H on the left of Figure 6 can be cancelled, ending up with the Givens pattern on the right of Figure 6? Maybe surprisingly, the answer to this question is no, since the pull-through operation may be badly determined. This means that in certain cases, the pullthrough of two Givens transformations which are equal up to the machine precision ≈ 10−16 may lead to relatively large diﬀerences in the pulled through Givens transformations. Stated in another way, inserting G does not always guarantee that (essentially) the computed H will come out at the other side. The solution to this problem is to use the information of both Givens transformation G and H, and to perform an explicit refactorization procedure of the

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unitary matrix Q in the left-hand side of Figure 6. In other words, we are given the matrix Q and our aim is to refactorize this matrix into a new Givens product having the form of the (unknown) right-hand side of Figure 6. Let us assume that r = 3, and let us denote the required refactorization as ˜ ˜ ˜˜ , ˜ 1,2 G ˜ 2,3 G ˜ 3,4 G ˜ 4,5 G ˜ 2,3 G Q=G 3,4

(20)

˜˜ ˜ i+1,i+2 and G where we have to search each of the Givens transformations G i,i+1 , i = 1, . . . , 3; cf. Figure 6. To obtain the refactorization (20) in a stable way, we will determine the ˜˜ ˜ i+1,i+2 and G subsequent Givens transformations G i,i+1 , i = 1, . . . , 3 as illustrated in Figures 8 and 9.

=

˜ H from a least Figure 8. Determine the ﬁrst Givens transformation G 2,3 squares condition.

1 =

˜˜ H . Figure 9. Determine the second Givens transformation G 1,2 Let us comment on these ﬁgures. The ﬁrst step is to note that the multiplica˜H tion of the matrix Q on the left with the Givens transformation G 2,3 should create a zero in the (3, 1) element as well as in the (2, 5) element of Q, i.e.,

q2,1 q2,5 × 0 ˜H G , (21) = 2,3 q3,1 q3,5 0 × ˜ H by using a pivoting see Figure 8. In practice, it is advisable to determine G 2,3 or least-squares strategy based on these two requirements. Let us then update ˜H Q := G 2,3 Q. The second step is to note that the multiplication of Q on the left with the ˜ ˜ H should create a zero in the (2, 1) element of Q, i.e., Givens transformation G 1,2

q1,1 × ˜ H ˜ , (22) G1,2 = 0 q2,1

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˜˜ H from this equation. We see Figure 9. Note that we can obviously determine G 1,2 ˜ ˜ H Q. can then update Q := G 1,2 The ﬁrst column of Q has then been brought completely into upper triangular form and hence, due to the unitarity, also the ﬁrst row of this matrix will vanish: see the right-hand side of Figure 9. Since the situation at the end of Figure 9 is similar to the one we started from in Figure 8, but with smaller dimension of the problem, the determination of the next Givens transformations of the refactorization (20) is not shown anymore. A suited use of the pull-through lemma enables the above refactorization scheme to be carried out in O(r) operations, rather than the O(r2 ) suggested in the above ﬁgures. The clue to this speed-up is to represent the matrix Q by its Givens product representation rather than expanding it in full form. The initial Givens product representation of Q is provided by the left-hand side of Figure 6. ˜˜ H ˜H For each of the updates Q := G i+1,i+2 Q and Q := Gi,i+1 Q, we then update the Givens product representation of Q by applying the pull-through lemma maximally two times in the upward direction; note that this requires only O(1) operations. Moreover, it turns out that in this way, the required elements q2,1 , q3,1 in (21) and q1,1 , q2,1 in (22) can be computed from the Givens product representation of Q using only O(1) operations. Applying this method for all the Givens transforma˜ ˜ i+1,i+2 and G ˜ i,i+1 , i = 1, . . . , r, reveals that the total complexity will be tions G O(r) operations. To complete the above description we should then show that also the elements q2,r+2 , q3,r+2 , . . . in (21) during this and the next steps of the refactorization process can be computed in O(r) operations. But since these elements all belong to the rightmost column of Q, this can be realized by just precomputing the latter column in its full form, and next updating it during the refactorization process. It is easy to see that the total cost of this process is O(r) operations as well. Summarized, we have described now how to implement the basic chasing step in Figure 5 in a practical way, using not more than O(r) operations. Applying this scheme for i = r + 1, . . . , n − r − 1, we can obtain the subsequent Givens transformations G := Gi,i+1 of (12) using a total number of O(rn) operations. To complete the description of the QR-factorization in the Cauchy-like case, we should still explain how to determine the ﬁrst and last r Givens transformations Gi,i+1 in (12), and how to update the representation under the action of these operations. First, let us consider the computation of the ﬁrst r Givens transformations Gi,i+1 , i = 1, . . . , r. This can be done by just specializing the general scheme of Section 3 in a straightforward way. Hence, it will suﬃce if we can show how the representation of the matrix S should be updated under the inﬂuence of a similarity transform with these operations. This is done in Figure 10. Let us comment on this ﬁgure. First, we embed the given Givens product representation of size n − 1, in a larger n by n representation. This can be achieved by pulling the two original branches apart, as done in Figure 10(a), and inserting

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(a)

(b)

(c)

Figure 10. Absorbing the ﬁrst Givens transformations into the representation. some identity Givens transformations between them, as indicated by the dashed line segments in Figure 10(a). Next, we multiply with the next unitary diagonal element y1,1 , indicated by the fat dot, and with the ﬁrst r Givens transformations Gi,i+1 on the left, GH i,i+1 on the right, indicated by the fat line segments in Figure 10(b). One can now immediately absorb the rightmost added Givens transformations into the representation, and place them on the position of the dashed lines, without any actual computation: see Figure 10(c). Having done this, one should apply the pull-through lemma r × r = r2 in the downward direction to bring the leftmost added Givens transformations downwards, so that they ultimately appear on the positions of the remaining set of dashed lines in Figure 10(c). Clearly, the complexity of this entire process is only O(r2 ) operations, which can be considered as negligible w.r.t. the other parts of the algorithm. Having done these operations, the Givens product representation will be brought back into its usual form, as in Figure 3. We move now to the absorption of the last r Givens transformations Gi,i+1 , GH i,i+1 into the representation, i = n − r, . . . , n − 1. This can be done even simpler, using straightforward pull-through operations as indicated in Figure 11. We wish to point out here the striking fact that absorbing the ﬁrst and last r Givens transformations into the representation both requires about O(r2 ) operations, and essentially the same number of pull-through applications. Finally, to conclude the description of the QR-factorization algorithm in the Cauchy-like case, we should still explain how each time the linear system is solved to compute the last column of the Hessenberg matrix H, as in (16). Since this linear system contains as coeﬃcient matrix S − zn,n I = QH Dy Q − zn,n I, with in general zn,n = 0, one can not proceed here with the Givens product representation anymore. Instead, we suggest to transform the Givens product representation ﬁrst into a Givens-weight representation, using the O(r2 n) transformation algorithm

QR-factorization of Displacement Structured Matrices

(a)

247

(b)

Figure 11. Absorbing the ﬁnal Givens transformations into the representation. described in [5]. Once the Givens-weight representation has been computed, the diagonal correction term −zn I is easily added to the matrix, and one can then proceed by the standard Givens-weight solvers, including also O(r2 n) operations [4].

5. The Vandermonde-like case In this section we brieﬂy consider the QR-factorization in the case of a Vandermonde-like matrix A. Denoting the upshift matrix

0 In−1 Zα = , (23) α 0 for any α ∈ C, and with In−1 the identity matrix of size n − 1, we will work with the Vandermonde-like matrices described by the displacement equation Z1 A − ADz = Rk r,

(24)

where Z1 is the circulant upshift matrix, having α = 1, and where Dz ∈ Cn×n is a diagonal matrix. We also assume that the displacement operator (24) is invertible and, therefore, that the spectra of Z1 and Dz do not intersect each other. The diagonal elements zi are sometimes called the interpolation nodes of the Vandermondelike matrix. The displacement rank r is assumed to be known; note that r = 1 in case of a classical (transposed) Vandermonde matrix A = [zji ]i,j=0,...,n−1 . It should be observed that the displacement equation (24) does not allow the general scheme for QR-factorization in Section 3 to be applied, since the ﬁrst coeﬃcient matrix of the displacement equation violates the requirement of being upper triangular. The solution consists in working instead with the matrix A˜ := F H A, where F is the Fourier matrix. From the well-known spectral decomposition Z1 = F Dy F H ,

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see, e.g., [13], the equation (24) transforms then to 6 r, ˜ z = Rk Dy A˜ − AD

(25)

where Dy is now a unitary diagonal matrix containing roots of unity. Clearly, the problem of QR-factorization has been reduced now to the Cauchy-like case described in the previous section. The total complexity of the QR-factorization remains O(r2 n) operations.

6. The Toeplitz-like case In this section we consider the case where A has Toeplitz-like structure. A ﬁrst idea could be to use the displacement equation Z1 A − AZ0 = Rk r, where we use the notations of (23) above. The displacement rank r is again assumed to be known; note that r = 2 in case of a classical Toeplitz matrix A = [ti−j ]i,j=1,...,n . Using again a premultiplication with the Fourier matrix F , the above problem of QR-factorization can be reduced immediately to the form 6 r, ˜ 0 = Rk Dy A˜ − AZ where Dy is again a unitary diagonal matrix containing roots of unity, and where we deﬁned the updated matrix A˜ := F H A. It can be observed now that in this case, the coeﬃcient matrix Z := Z0 is strictly upper triangular, i.e., zi,i = 0 for each i. This has two important consequences: • also the main diagonal is involved into the lower semiseparable structure; • the coeﬃcient matrix in (16) is the matrix QH Y Q itself, i.e., there is no diagonal correction which has to be added to this coeﬃcient matrix. This last observation will allow for an O(rn2 ) solution of the QR-factorization problem; this should be compared to the O(r2 n2 ) solution in case of Cauchy-like matrices, see further. Let us illustrate here the corresponding unitary rank structured matrix S: see Figure 12. The diﬀerence for the corresponding unitary representation is now that the left branch of the representation of the unitary rank structured matrix S is slightly thicker than the right one, by the fact that the diagonal is part of the rank structure. This has the important consequence that the left and right Givens transformations G and GH with G := Gi,i+1 do not inﬂuence the same structure block anymore, i.e., the algorithm has obtained a well-determined ﬂow direction. In practice, this amounts to the fact that the Givens transformation G := Gi,i+1 , after pulling it through the representation, arrives on the right of the right branch of the representation, one position lower than it was originally standing. This pulled-through

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Rk 3 Rk 3 Rk 3 Rk 3 Rk 3 Rk 3 Rk 3 Rk 3 Rk 3 Rk 3 Rk 3 Rk 3

Figure 12. Rank structure in the Toeplitz-like case; we have here r = 3. Givens transformation determines then immediately the Givens transformation Gi+1,i+2 to be applied during the next step, and so on. See Figure 13.

G

H

Figure 13. The ﬁgure shows the ith chasing step in the Toeplitz-like case, i ∈ {3, . . . , n − 4}. Note that the original Givens transformation G = Gi,i+1 is pulled through to the right, leading to a new Givens transformation H = Gi+1,i+2 . The latter serves then as input for the (i + 1)th chasing step. In the ﬁgure we assume that r = 3. Applying the above scheme for i = r, . . . , n − r − 1, one can obtain the subsequent Givens transformations Gi,i+1 in (12). We should then show how to obtain the ﬁrst r and last r − 1 Givens transformations Gi,i+1 . This can be done by just specializing the general scheme of Section 3 in a straightforward way. Hence, it will suﬃce if we can show how the representation of the matrix S should be updated under the action of the ﬁrst r − 1 operations on the left and the ﬁrst r operations on the right. But this is completely similar to the Cauchy-like case: see Figure 14. Concerning this ﬁgure, note that the only diﬀerence w.r.t. the Cauchy-like case is that the symmetry between the number of Givens transformations on the rows and columns has somewhat changed, due to the asymmetry of the two branches of the representation. In case of Figure 14, this implies the absorption

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Figure 14. Absorbing the ﬁrst Givens transformations into the representation in the Toeplitz-like case. of the ﬁrst three Givens transformations on the right, but only of the ﬁrst two Givens transformations on the left. Indeed, this reﬂects the fact that starting from the third Givens transformation G3,4 on the left, one can start with the chasing scheme of Figure 13. In a similar way, one can treat the absorption of the last r Givens transformations on the rows and the last r − 1 Givens transformations on the columns, where now also the symmetry between the number of Givens transformations on the rows and columns has changed, in an obvious way. The details are straightforward.

7. Numerical experiments In this section we discuss the practical behavior of the proposed algorithms for the recursive QR-factorization of displacement structured matrices and report on the results of some numerical experiments. The algorithm for the Cauchy-like case described in Section 4 has been implemented in Matlab and then tested on both random and speciﬁc input matrices. In each experiment we measured the accuracy of the computed QR factorization of the input matrix C by evaluating the errors ||Q · QH − I||F , ||C − Q · R||F and ||tril(QH · C, −1)||F , where tril(A, −1) denotes the strictly lower triangular part of the matrix A and ||·||F is the Frobenius matrix norm. Our program also returns as output the estimates of the conditioning of C and of the maximal conditioning of the lower-left submatrices of C employed in the recursive factorization process. In our ﬁrst test session the input matrices were generated as follows. We started from a Toeplitz matrix T ∈ Cn×n determined by the entries aj , j = 1, . . . , 2n − 1, with aj , j = 1, . . . , n corresponding to the ﬁrst row and aj , j = n, . . . , 2n − 1 corresponding to the ﬁrst column of T . Applying a similarity transform C := F T F H , with F the Fourier matrix of size n, transforms the Toeplitz matrix T into a Cauchy-like matrix C satisfying Dy C −CDz = Rk 2, where Dy and Dz contain interlaced roots of unity; this is the same construction as we used in

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S. Delvaux, L. Gemignani and M. Van Barel (a)

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dealt with by using any standard stable routine for updating a QR-factorization under the inﬂuence of a low rank correction term. Figures 16 and 17 report the results for α = 0.48 and α = 0.45, respectively. It is seen that the initial randomization improves substantially the accuracy of the method. Moreover, Figure 17 seems to indicate that the accuracy is inﬂuenced by the conditioning of the matrices involved in the recursive factorization process. This claim sounds reasonable since from the classical perturbation analysis for the QR factorization of a nonsingular matrix C it follows that the magnitude of the perturbation ΔQ of the Q-factor of C under a small perturbation ΔC of C can be ampliﬁed by a factor depending on the conditioning of C [14].

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However, we want to conclude by stressing that the mechanism of error ampliﬁcation in the chasing-based procedures for the QR factorization of displacement structured matrices in not yet completely clear. This is speciﬁcally true for not diagonal displacement operators as those occurring with Toeplitz-like structures. Some very preliminary numerical experiments with Toeplitz matrices suggest that in that case the accuracy of computed results can be seriously worse than it is predicted by the conditioning of the problem. At this time we have no analysis or informal explanation for this phenomena and more research is still required for the design of numerically robust algorithms. Acknowledgment The ﬁrst author would like to thank Prof. Gemignani for inviting him in September 2005 at the University of Pisa, Italy, where this joint work has started.

References [1] A.W. Bojanczyk, R.P. Brent, and F.R. de Hoog. QR factorization of Toeplitz matrices. Numerische Mathematik, 49:81–94, 1986. [2] S. Delvaux, L. Gemignani, and M. Van Barel. Fast QR-factorization of Cauchy-like matrices. Linear algebra and its Applications, 428, 2008. [3] S. Delvaux and M. Van Barel. A Givens-weight representation for rank structured matrices. SIAM Journal on Matrix Analysis and its Applications, 29(4):1147–1170, 2007. [4] S. Delvaux and M. Van Barel. A QR-based solver for rank structured matrices. SIAM Journal on Matrix Analysis and its Applications, 2008. To appear. [5] S. Delvaux and M. Van Barel. Unitary rank structured matrices. Journal of Computational and Applied Mathematics, 215(1):49–78, 2008. [6] P. Dewilde and A.-J. van der Veen. Time-varying systems and computations. Kluwer Academic Publishers, Boston, June 1998. [7] D. Fasino and L. Gemignani. A Lanczos type algorithm for the QR-factorization of regular Cauchy matrices. Numerical Linear Algebra with Applications, 9:305–319, 2002. [8] L. Gemignani, D.A. Bini, Y. Eidelman, and I.C. Gohberg. Fast QR eigenvalue algorithms for Hessenberg matrices which are rank-one perturbations of unitary matrices. SIAM Journal on Matrix Analysis and its Applications, 29, 2007. [9] G. Heinig and A.W. Bojanczyk. Transformation techniques for Toeplitz and Toeplitzplus-Hankel matrices I. Transformations. Linear Algebra and its Applications, 254:193–226, 1997. [10] G. Heinig and A.W. Bojanczyk. Transformation techniques for Toeplitz and Toeplitzplus-Hankel matrices II. Algorithms. Linear Algebra and its Applications, 278:11–36, 1998. [11] G. Heinig and K. Rost. Representations of Toeplitz-plus-Hankel matrices using the trigonometric transformations with application to fast matrix-vector multiplication. Linear Algebra and its Applications, 275-276:225–248, 1998.

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[12] T. Kailath and A.H. Sayed, editors. Fast reliable algorithms for matrices with structure. SIAM, Philadelphia, PA, USA, May 1999. [13] V.Y. Pan. Structured matrices and polynomials. Birkh¨ auser Springer, 2001. [14] X.W. Chang and C.C Paige and G.W. Stewart. Perturbation analyses for the QR factorization. SIAM Journal on Matrix Analysis and its Applications, 18(3):775–791, 1997. [15] M. Van Barel, D. Fasino, L. Gemignani, and N. Mastronardi. Orthogonal rational functions and structured matrices. SIAM Journal on Matrix Analysis and its Applications, 26(3):810–829, 2005. [16] C.F. Van Loan. Computational Frameworks for the Fast Fourier Transform. Frontiers in Applied Mathematics. SIAM, 1992. Steven Delvaux Department of Mathematics Katholieke Universiteit Leuven Celestijnenlaan 200B B-3001 Leuven (Heverlee), Belgium e-mail: [email protected] Luca Gemignani Dipartimento di Matematica Universit` a di Pisa Largo Bruno Pontecorvo 5 I-56127 Pisa, Italy e-mail: [email protected] Marc Van Barel Department of Computer Science Katholieke Universiteit Leuven Celestijnenlaan 200A B-3001 Leuven (Heverlee), Belgium e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 199, 255–288 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Bezoutians Applied to Least Squares Approximation of Rational Functions Sven Feldmann To Georg Heinig

Abstract. A projection method to reduce large scale discrete systems which has been introduced in [12, 21] will be generalized to continues systems without to transform it bilinear. To achieve that goal depending on an algebraic curve γ ⊂ C and a rational function h ∈ C(z) a non negative function F : Cm → R is introduced whose minimizer provides an approximant of degree m. Special cases are obtained via speciﬁcation of γ and h. Mathematics Subject Classiﬁcation (2000). 15A24; 41A20. Keywords. Bezoutian, matrix equation, model reduction, stability.

1. Introduction The paper concerns an approximation problem. Let γ ⊂ C be a curve which divides the complex plane C in two parts C and Cr , and h ∈ C(z) be a stable strictly proper rational function, that means all it poles, denoted with σ(h), are located in C and limz→∞ h(z) = 0. We are interested in the approximation of h by a low (Mc-Millan) degree strictly proper rational function hred such that along γ the least squares distance between h and hred is satisfying and hred is again stable. As usual the degree of h is deﬁned by the degree of q ∈ C[z] where p/q is a coprime fraction representation of h. That approximation problem admits a system theoretical interpretation via deﬁnition of h by the transfer function hΣ (z) := C(zI − A)−1 B of the systems xk+1 yk

= Axk + Buk , x0 = 0 = Cxk

x(t) ˙ = Ax(t) + Bu(t) , x(0) = 0 y(t) = Cx(t)

Σ := (A, B, C) ∈ Rn×n × Rn×1 × R1×n

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which transform the inputs uk := Cu Aku Bu ,

u(t) := Cu exp(Au t)Bu ,

Σu := (Au , Bu , Cu ) ∈ Rn×n × Rn×1 × R1×n ,

σ(Au ) ∩ σ(A) = ∅

to the outputs )u + CAk B, ) yk = Cu Aku B

)u + C exp(At)B ) y(t) = Cu exp(Au t)B

)u := hΣ (Au )Bu , B

) := hΣu (A)B. B ∞ k As usual exp(A) and h(A) are deﬁned, respectively, by k=0 Ak! and p(A)q(A)−1 . Among other things, large n makes the computation of the outputs infeasible in acceptable time. That is one reason why one is interested in the replacement of Σ by a smaller triple Σred which generates systems that have similar transfer behavior. To explain the term ‘similar transfer behavior’ we restrict ourself to γ ∈ {T, iR},

T := {z ∈ C : |z| = 1},

iR := {z ∈ C : (z) = 0}.

Here, (z) designates the real part of the complex number z. For such γ the left-hand parts C are the unit disc and the left half-plane D := {z ∈ C : |z| < 1},

H− := {z ∈ C : (z) < 0}.

Thus, the matrices A and exp(At) tend to zero as k, t → ∞ such that for large )u and enough k and t the quantities yk and y(t) are in essential equal to Cu Aku B ) Cu exp(Au t)Bu . Consequently, two stable systems Σ and Σred possess similar transfer behavior relative to the input signal class k

Sγ := {Σu : σ(Au ) ⊂ γ} if for all Σu ∈ Sγ the number d(Au ) := hΣ (Au ) − hΣred (Au )F is small. For Σu ∈ Sγ the smallness of hΣ − hΣred γ2 leads to7the smallness of d(Au ). Here, n 2 .F denotes the Frobenius matrix norm AF := i,j=1 |aij | , [[18], p. 56] and .γ2 the L2 -norm along γ, deﬁned by 5 γ γ f γ2 := f |f 2 , f |g2 := γ := {w(t) : t ∈ [a, b]},

1 2π

3b

f (w(t))g(w(t))|w(t)|dt ˙ w : [a, b] ⊆ R → C. a

Hence, for a stable Σ, stimulated by elements of Sγ , the reduction problem is equivalent to approximate hΣ along γ by a stable strictly proper low degree rational function hred . The desired reduced system Σred is provided by a realization of hred . For γ ∈ {T, iR} the balanced truncation method [[35], Section 21.8, Chapter 7] represents a prominent reduction procedure. An overview with respect to the huge number of papers which deal with approximation problems of this nature oﬀer [4, 5]. The optimal γ-approximation problem of order m consists in ﬁnding γ := {g ∈ C(z) : σ(g) ∈ C , deg g = m} hopt ∈ Rm

such that γ

γ

γ h − hopt 2 = min{h − g2 : g ∈ Rm }.

(1.1)

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γ does not not form a subspace of {g ∈ C(z) : gγ2 < ∞} the determinaSince Rm tion of an hopt becomes diﬃcult. Via the speciﬁcation γ = iR the so-called optimal H2 -model reduction problem is obtained. There exists a lot of corresponding investigations, consult for instance [30, 32, 34]. We do not determine hopt but we try to provide an hred such that

h − hred γ2 ≈ h − hopt γ2 . To achieve that goal we associate h with two ordered polynomial sets A := {a0 , . . . , an−1 }, R := {r0 , . . . , rn−1 } and consider as approximation candidates the rational functions m−1 am (z) − k=0 xk ak (z) , x ∈ Cm , ak ∈ A , 0 < m < n. hx (z) := m−1 z m − k=0 xk z k To ﬁnd appropriate weight vectors x, depending on R and γ we deﬁne a non negative function F : Cm → R and derive the representation F (x)2 = row(x∗ , −1)Gm+1 col(x, −1),

Gm ∈ Cm×m .

(1.2)

It turns out that the minimizer ξ of F generates an hξ having all desired properties of hred and that G := Gn admits the factorization G = R∗ Bγ−1 (q)R.

(1.3)

We call G the Gramian of h with respect to γ. Due to Bγ (q) is representing the Bezoutian of q with respect to γ, that factorization brings together the Bezout concept with rational approximation problems, admits a natural generalization of the Bezout concept and justiﬁes paper’s title. The context to [12, 21] is established by the possibility to compute ξ via a projection method which has been introduced there. To analyze the properties of hξ the class of admissible curves will be restricted to the class of algebraic curves γ := {z ∈ C : pγ (z, z) = 0} .

(1.4)

Here, pγ is a bivariate polynomial with Hermitian coeﬃcient matrix Γ: T pγ (x, y) := ψν+1 (x)Γψν+1 (y), 0 ≤ γνν , Γ := [γij ]νi,j=0 , ψn (x) := row(xj )n−1 j=0 .

That restriction allows us to associate h and γ with the Hermitian matrix L deﬁned by L := Lγ (Cq , M ) ∈ Cn×n , M := G−1 ν Lγ (A, X) : Cn×n → Cn×n , Lγ (A, X) := i,j=0 γij Ai XA∗j , A ∈ Cn×n . Remember, that for A, B ∈ Cn×n , γ = T and ν = 1 the matrix equation Lγ (A, X) = B

(1.5)

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is said to be a Stein and for γ = iR a Lyapunov equation. We pay our main attention to the case ν = 1. Then the curvature of γ is constant such that C is a disc or a half-plane and non-positive L implies σ(hξ ) ⊂ C := C ∪ γ. Furthermore, for positive

√ Δ := − det Γ the desired non-positivity is guaranteed by the existence of w ∈ Cn satisfying L = −Δww∗ .

(1.6)

Our construction of w is derived from the special cases γ ∈ {T, iR}. In particular, in the unit circle case Parseval’s identity states the equivalence between the deﬁnition of the denominator polynomial Q of hξ provided here, deﬁnition (10) of Qn in [12] and deﬁnition (3.4) of χLS in [21]. Numerical experiences show that a small distance between σ(h) and γ is preserved by σ(hξ ) and that σ(hξ ) possesses an element nearby to α ∈ σ(h) supposed the associated partial fraction coeﬃcient is large. A heuristical discussion of that observations took place in [13]. The theoretical part of the present paper ﬁnishes with a mathematical foundation. Their dependencies determine the paper’s structure. In Section 2 we deﬁne F and G precisely and deduce representation (1.2). Afterwards, an estimate of the distance between h and hx directly proportional to F (x) is derived such that the deﬁnition of x via its minimizer ξ becomes natural. Due to factorization (1.3) Section 3 is dedicated to Bezoutians. In Section 4 we prove σ(hξ ) ⊂ C by exploitation of the matrix equation (1.6) and give an explicit construction of w. The paper’s main theorem (Theorem 4.4) combines the pole separation result with the distance estimate along γ such that we arrive at a generalization of [[12], Theorem 2] and [[21], Theorem 3.1.]. As mentioned above, the construction receipt for w is derived from the special cases γ ∈ {T, iR}. Therefore, Section 5 and 6 deal with that cases. Section 7 focuses on the computation of hξ starting from the knowledge of a minimal realization of h. That happens by identiﬁcation of G as the observability Gramian of the controllability realization of h with respect to γ. In the extreme situation h = 1/q the set A becomes a singleton consisting only of the zero polynomial such that as approximant the zero function is only in question. To overcome that collapse, in Section 8 a second minimization step is concatenated to the minimization of F which deﬁnes the numerator of hξ more sensitive such that for the resulting rational function hηξ the relation γ

h − hηξ γ2 ≈ h − hopt 2

is achieved. In particular by introduction of the mirror image of σ(Q) with respect to γ the numerator polynomial P of hηξ can be obtained as solution to an interpolation problem. In Section 9 the limit behavior of σ(hξ ) is studied when some pole of h moves to the border γ or a partial fraction coeﬃcient tends to inﬁnity. The main tool to analyze such limit processes are the generating functions of Bezoutians. Since there exist bilinear transformations between D and H− the

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statements with respect to D can be translated in corresponding statements with respect to H− . In Section 10 we sketch that approach. Finally, in Section 11 we illustrate our theoretical results by means of numerical examples. The both last examples touch the concept of positive realness. It turns out that the direct as well as the transformation approach do not preserve the underlying mapping property. In a forthcoming paper that problem will be studied.

2. The construction of hξ Depending on a diﬀerentiable curve γ, a positive measure μ acting on γ and polynomials rj ∈ R ⊂ Cn [z] := {p ∈ C[z] : deg p < n} we deﬁne F : Cm → R, m < n, according to px (z) := rm (z) − m−1 F (x) := px γdμ , j=0 xj rj (z), 3 γ γ 12 γ f dμ := f | f dμ , f | gdμ := γ f g dμ. To adapt F to our approximation problem relative to p/q := h, p, q ∈ C[z], the sets A and R are speciﬁed by the solutions to z j p = aj q + rj , rj ∈ Cn [z], aj ∈ Cj [z], j = 0, . . . , n − 1, C0 [z] := {0}. We call rj the (j + 1)th residue of p with respect to q. Cayley-Hamilton’s Theorem [[35], p. 21] shows the coincidence of the coeﬃcient vector rj of rj regarded as an element of Cn with the j th column of p(Cq ). Thus, the deﬁnition R := r0 . . . rn−1 ∈ Cn×n leads to R = p(Cq ). Due to r0 = p the ﬁrst column of R coincides with the coeﬃcient vector p of p. Moreover, p(Cq ) = K (Cq , p),

K (A, B) := Kn (A, B),

K (A, B) := row(Aj B) −1 j=0 .

The measure μ is speciﬁed by dμ(t) :=

|w(t)| ˙ d d γ := {w(t) : t ∈ [a, b]}, dt, w˙ := (w) + i !(w). w : [a, b] ⊆ R → C, 2π|q(w(t))|2 dt dt

As approximation candidates now the elements hx of the rational function set 8 2 m−1 P (z) := am (z) − k=0 xk ak (z), m x ∈ C Rm (h) := P : m−1 Q Q(z) := z m − k=0 xk z k , are in question. Note that for fk := rk /q we have fk (z) = z k h(z) − ak (z) = CAk (zI − A)−1 B, a0 (z) ≡ 0, ak+1 (z) = ki=0 hk−i z i , hk := CAk B. Here, (A, B, C) represents a realization of h as for example the controllability real- T ization Σco := (Cq , e1 , h∗ ) [[14], p. 288] or the controller realization CqT , en , q−1 n p

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where (regard blank spaces as ﬁlled with zeros) ⎡ ⎤ −q0 /qn p = ψn p ⎢ 1 −q1 /qn ⎥ ⎢ ⎥ q = ψn+1 q , Cq := ⎢ ⎥ .. .. ⎣ ⎦ . . (2.1) h := col(hi )n−1 i=0 1 −qn−1 /qn T T e1 := 1 0 . . . 0 , en := 0 . . . 0 1 . k The equation ak+1 (z) = i=0 hk−i z i and the deﬁnition of P admit to express the coeﬃcient vector ω of P according to ⎤ ⎡ h0 . . . hm−1

⎥ −col(xi )m−1 ⎢ .. .. i=1 ω=⎣ . (2.2) ⎦ . . 1 h0 Therefore, Rm (h) forms the set of all strictly proper rational functions of degree m whose ﬁrst m Laurent coeﬃcients coincide with (hk )m−1 k=0 . For discrete systems that property means the coincidence of the unit impulse responses up to the length m. If one renounces to preserve that system property, the gained freedom to choose P can be used to proceed with a second minimization step after minimization of F . Section 8 focuses on the corresponding details. To estimate h− hx γ2 we introduce min(f ) := min{|f (w)| : w ∈ γ}, γ

max(f ) := max{|f (w)| : w ∈ γ}. γ

Proposition 2.1. Supposed σ(Q) ∩ γ = ∅ the estimate h − hx γ2 ≤

F (x) minγ (Q)

holds.

Proof. Replacement of rj by z j p − aj q yields m−1 m−1 px = rm − j=0 xj rj = (z m p− am q) − j=0 xj z j p − aj q m−1 m−1 = z m − j=0 xj z j p − am − j=0 xj aj q = Qp − P q. γ γ Thus h − hx γ2 = px /(Qq)2 ≤ maxγ (Q−1 ) px /q2 =

1 minγ (Q)

γ

px dμ .

Consequently, to get along γ a small distance between h and hx , the choice of x via the minimizer ξ of F is natural. To determine ξ note that 9 :γ m−1 m−1 F (x)2 = rm − j=0 xj rj rm − j=0 xj rj dμ (m

∗ ' x γ = x −1 ri | rj dμ . −1 i,j=0 Using the abbreviations m−1

γ

G := Gn , Gm := [gij ]i,j=0 , gij := ri | rj dμ the factorization (1.2) is obtained. We call G the Gramian of h with respect to γ. In addition we set m gm := col(gim )m−1 i=0 ∈ C .

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261

If one deﬁnes an ∈ Cn [z] and rn ∈ Cn+1 [z] via z n p = an q + rn then the vector gn is deﬁned and q admits the representation q(z) = qn (z n − ψn (z)G−1 gn ). Proposition 2.2. Supposed p and q are coprime F achieves its minimum in G−1 m gm . Thus, we set (2.3) Q(z) := z m − ψm (z)ξ, ξ := G−1 m gm . Proof. Obviously,

' (n−1 G = R∗ z i | z j γdμ R

(2.4)

i,j=0

where the positivity of the middle factor is a consequence of the positivity of μ. Since p and q are coprime, together with R = p(Cq ) the invertibility of R follows. Thus G is positive. We proceed with the principal axis transformation of Gm . Let ∗ Gm = U diag(dk )m k=1 U

where U is unitary and all dk are positive. With x ˆ := U x and gˆm := U gm we get m xk , dk , gˆkm ), f (z, d, g) := d|z|2 − 2(gz). F (x)2 = gmm + k=1 f (ˆ Therefore, the minimum of F 2 is equal to m gmm + k=1 minz∈C f (z, dk , gˆkm ). The condition fx = fy = 0, z = x + iy, leads to 2dx − 2(g) = 0,

2dy − 2!(g) = 0.

Hence, x ˆk = gˆkm /dk that means G−1 m gm represents the only one stationary point of 2 F . Since the (2 × 2)-matrix of the second derivations of f is equal to diag(dk , dk ) and dk is positive, F attains its minimum there.

3. Bezoutians We introduce now the Bezout concept for q with respect to γ. The classical Bezoutians are special Hermitian matrices that were introduced to study the root location problem of polynomials. The naming of Bezoutians goes back to J. Sylvester [31]. Historical remarks can be found in [25, 26, 33]. In view of (1.3) and (2.4) we set

' <γ (n−1 −1 ; i z j dμ z . Bγ (q) := i,j=0

In Section 5 and 6 for γ ∈ {T, iR} and σ(q) ⊂ C the coincidence of Bγ (q) with the classical Bezoutians Bd (q) := H (q1 , . . . , qn )H (q1 , . . . , qn ) − H (qn−1 , . . . , q0 )H (qn−1 , . . . , q0 ) q0 , . . . , qn−1 )H (q1 , . . . , qn ) − T (q0 , . . . , qn−1 )H ( q1 , . . . , qn )) Bc (q) := I± (T ( (3.1)

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S. Feldmann

q := I± q and will be proved, a fact that justiﬁes to refer Bγ (q) as Bezoutian. Here, % % x1 ... xn & % x1 & & 1 −1 . . . I± := , T (x1 , . . . , xn ) := .. . . . . . . , H (x1 , . . . , xn ) := .. . . xn xn ... x1 . The right-hand side of (3.1) is said to be the Gohberg-Semencul representation of Bγ (q) [[14], p. 206]. In view of [[26], Theorem XVa], [[24], Theorem 4.4], [[15], Theorem B2], [[14], Theorem 9.1.5.] the relations σ(q) ⊂ D ⇔ 0 < Bd (q), σ(q) ⊂ C \ D ⇔ Bd (q) < 0 σ(q) ⊂ H− ⇔ 0 < Bc (q), σ(q) ⊂ C \ H− ⇔ Bc (q) < 0

(3.2)

hold. To ﬁnd the vector w satisfying L = −Δww∗ it is worth to consider appropriate representations of the bivariate polynomial Bγ (q, x, y) := ψn (x)Bγ (q)ψnT (y). According to [[14], Theorem 9.1.5, p. 256] we have Bd (q, x, y) =

) q(x)) q (y) − q(x)q(y) , 1 − xy

q)(z) := q(z

−1

n

)z ,

Bc (q, x, y) =

q (z) := q(−z),

q(x)q(y) − q(x) q (y) x+y

q(z) := q

∗

(3.3)

T ψn+1 (z).

The ﬁrst factors of the products arising in the numerators provide now w. Indeed, w = R−1 v where for γ = T the polynomial v =: ψn v is the ﬁrst residue of ) q q . The with respect to q and for γ = iR the ﬁrst residue of q with respect to representation (3.3) of Bγ (q, x, y) is said to be the generating function of Bγ (q).

4. The matrix L for ν = 1 and indeﬁnite Γ We specify γ by (1.4) and set C := {z ∈ C : pγ (z, z) < 0}. Our proof of σ(hξ ) ⊂ C needs the following representation of Q. Remember that M is deﬁned by G−1 . Proposition 4.1. Let M be partitioned according to [Mij ]2i,j=1 , M11 ∈ Cm×m . Then M22 is invertible and Q admits the representation n −1 e1 ∈ Cn−m , mj := ψn M ej ∈ Cn [z]. Q(z) = j=m+1 ζj mj (z), ζ := M22 Proof. Since G is positive, the same holds for M . Consequently, M22 is invertible. Remembering the equation −1 G−1 11 G12 = −M12 M22 ,

2

[Gij ]i,j=1 := G,

M11 , G11 ∈ Cm×m

with G11 = Gm , gm = G12 e1 and ξ = G−1 m gm one concludes −1 e1 Q(z) = z m − ψm (z)ξ = z m + ψm (z)M12 M22

= z m + ψm (z)M12 ζ = ψn (z)col(Mi2 )2i=1 ζ =

n

j=m+1 ζj mj (z).

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263

We state and prove now our stability result assuming ν = 1 and the nonpositivity of L. As mentioned above, ν = 1 implies the coincidence of C with an open disc or an open half-plane. Proposition 4.2. For ν = 1 and L ≤ 0 we have σ(hξ ) ⊂ C . In the circle case hξ is stable. Proof. A not vanishing q(α) provides the inversion formula ⎡ 0 α0 ⎡ ⎤ ⎡ n−1 ⎤ qn . . . q1 α ⎢ 0 ⎢ . ⎥ ⎢ . ⎥ ψn (α) ⎢ .. (αI − Cq )−1 = ⎣ . .. ⎦ ⎣ .. ⎦ q(α) − ⎢ ⎢ ⎣ qn α0

. . . αn−2 .. .. . . .. . α0 0

⎤ ⎥ ⎥ ⎥. ⎥ ⎦

According to Proposition 4.1 the polynomial Q admits the representation Q = ψn τ,

τ := M ,

:= col(0, ζ).

Since deg Q = m the vector τ is of the form col(Q, 0), Q ∈ Cm+1 . Assume Q(α) = −∗ 0, q(α) = 0 and let ∈ Cn be deﬁned by := (αI − Cq ) . Then and τ are orthogonal: ∗ τ = ∗ (αI − Cq )−1 τ ⎤ ⎡ 0 α0 . . . αn−2 ⎡ ⎤⎡ n−1 ⎤ qn . . . q1 α .. ⎥

∗ ⎢

.. . Q 0 . ⎥ 0 ⎢ ⎥⎢ .. ⎥ Q(α) . ∗⎢ . ⎥ ⎢ . . = ⎣ . . ⎦⎣ . ⎦ q(α) − ζ ⎥ 0 = 0. ⎢ .. 0 ⎦ ⎣ . 0 α qn α 0 Together with the deﬁnition of the real numbers ϑ := ∗ L ≤ 0,

κ := ∗ M > 0,

λ := ∗ M > 0

and the equations ∗ Cq = α∗ − ∗ , τ = M we get ϑ = γ11 ∗ Cq M Cq∗ + γ10 ∗ Cq M + γ01 ∗ M Cq∗ + γ00 ∗ M = γ11 (α∗ − ∗ )M (α − ) + γ10 (α∗ − ∗ )M + γ01 ∗ M (α − ) + γ00 κ = γ11 (αακ − τ ∗ α − α∗ τ + λ) + γ10 (ακ − τ ∗ ) + γ01 (ακ − ∗ τ ) + γ00 κ = (γ11 αα + γ10 α + γ01 α + γ00 )κ + γ11 λ = pγ (α, α)κ + γ11 λ. Thus pγ (α, α) = (ϑ − γ11 λ)/κ ≤ 0 that means α ∈ C . For positive γ11 the inequality pγ (α, α) < 0 holds that means α ∈ C . In Example 2 we meet the situation σ(hξ ) ∩ γ = ∅ actually. Thus, without additional assumptions the statement of Proposition 4.2 cannot be strengthened. In particular, for the case where γ is a line and h := 1/q the localization of σ(Q) is a consequence of Fej´er’s convex hull theorem [[9], Thm. 10.2.2], [10]. In Section 6 for

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S. Feldmann

h = p/q and m := deg q − deg p we prove that Q becomes stable also for the halfplane case, and in [11] it will be shown that for m ∈ {deg q − deg p + 1, . . . , n − 1} the polynomial Q is generic stable. The proofs of Proposition 4.3 and 6.1 apply the bilinear transformation φ : C → C,

φ(z) :=

az + b , cz + d

ad − bc = 0

with appropriate parameters a, b, c, d ∈ C. The associated n-dimensional M¨ obius matrix Mφ is formed by the coeﬃcient vectors of {(az + b)j−1 (cz + d)n−j : j = 1, . . . , n}. For vanishing c we extent the coeﬃcient vector of (az + b)j dn−j to an element of Cn via a corresponding number of zeros such that Mφ becomes a right upper triangular matrix. If one sets pφ (z) := p(φ(z))(cz + d)n−1 ,

deg p < n

then the associated coeﬃcient vectors p and pφ are related by Mφ p = pφ . For the ' ( . 1 and I . transformations z −1 and −z the M¨ obius matrices read as J := .. ± 1 Thus ) q = Jq, ) q = Jq, q = I± q, q = I± q. For the parameters a = −1 and b = c = d = 1 the properties of M 5φ are described in [[27], p. 37]. Note, that for ν = 1 the quantity Δ is equal to |γ01 |2 − γ00 γ11 and is assumed to be positive. Proposition 4.3. Stable h implies the existence of w ∈ Cn with L = −Δww∗ . Proof. By assumption Δ is positive. Thus for positive γ11 , the curve γ represents a circle and for vanishing γ11 a line. Let w(t) := φ(z(t)) where φ(z) := az + b (c = 0, d = 1) and t ∈ I . Then w(t) is a parameterization of γ where a, b, z(t), I are deﬁned by 0 < γ11 0 = γ11

a Δ/γ11 1/γ10

b z(t) −γ01 /γ11 exp(it) −γ00 /(2γ10) it

I [0, 2π] R

Using the abbreviations γ := φ−1 (γ), B := Bγ (q), B := Bγ (qφ ) immediately by deﬁnition the identities B = |a|Mφ BMφ∗ ,

aCqφ Mφ = Mφ (Cq − bI)

follow. Combination of (1.3) and (1.5) with the commutator equation RCq = Cq R yields L = R−1 Lγ (Cq , B)R−∗ . Hence, setting w := R−1 Mφ−1 v it remains to compute the appropriate v.

Bezoutians Applied to Least Squares Approximation

265

In the circle case we have γ11 a = Δ. Thus Lγ (Cq , B) = γ11 ((Cq − bI)B(Cq − bI)∗ − a2 B) = Δa(Mφ−1 Cqφ Mφ BMφ∗ Cq∗φ Mφ−∗ − B) = ΔMφ−1 (Cqφ B Cq∗φ − B )Mφ−∗ . Due to γ = T and σ(q) ⊂ C ⇔ σ(qφ ) ⊂ D Proposition 5.2 states the existence of v such that Cqφ B Cq∗φ − B = −vv∗ . In the line case we have |a|−1 = Δ. Thus = γ10 (Cq − bI)B + B(Cq − bI)∗ γ01

Lγ (Cq , B)

= γ10 a Mφ−1 Cqφ Mφ B + BMφ∗ Cqφ Mφ−∗ aγ01 =1

=1

= ΔMφ−1 (Cqφ B + B Cq∗φ )Mφ−∗ . Due to γ = iR and σ(q) ⊂ C ⇔ σ(qφ ) ⊂ H− Proposition 6.2 states the existence of v such that Cqφ B + B Cq∗φ = −vv∗ . The combination of the Propositions 2.1, 4.2 and 4.3 generalizes [[12], Theorem 2] and [ [21], Theorem 3.1]. Theorem 4.4. In general we have σ(hξ ) ⊂ C . Moreover, for the circle case hξ is stable. Supposed σ(hξ ) ∩ γ = ∅ the approximant hξ satisﬁes γ

h − hξ 2 ≤ F (ξ)/ min(Q). γ

Proof. Proposition 2.1 provides the estimate and Proposition 4.3 states the as sumption of Proposition 4.2 which provides the location of σ(hξ ).

5. The unit circle case γ = T The entries of G read as T

gij = ri |rj dμ , f |gTdμ =

1 2π

3 2π 0

f (eit )g(eit )|q(eit )|−2 dt.

Moreover, for σ(q) ⊂ D the coincidence of BT (q) with the Toeplitz Bezoutian Bd (q) can be proved. The name Toeplitz Bezoutian is justiﬁed by the fact that Bd−1 (q) is a Toeplitz matrix. Matrices whose entries are constant along each diagonal are called Toeplitz matrices. To prove the asserted coincidence we use the semi-inﬁnite Hankel matrix H generated by 1/q: H := [hi+j ]∞,n−1 i,j=0 ,

h0 z −1 + h1 z −2 + h2 z −3 + · · · := q(z)−1 .

Matrices whose entries are constant along each anti-diagonal are called Hankel matrices. Proposition 5.1. σ(q) ⊂ D ⇔ Bd (q) = BT (q).

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S. Feldmann

Proof. Due to σ(q) ⊂ D the complex number sequence (hk )∞ k=0 belongs to 2 where as usual 2 denotes the normed vector space of all square-summable complex number sequences: 1/2 ε ∈ 2 ⇔ ε2 := |ε0 |2 + |ε1 |2 + · · · < ∞. The Schwarz inequality [[29], p. 77] provides the existence of H ∗ H. The Hankel structure of H implies the solution property of H ∗ H with respect to Cq∗ XCq − X = −en |qn |−2 e∗n .

(5.1)

By exploitation of the structure of Cq one shows that the product ∗ q)(Cq∗ )−1 q−1 n K (Cq , en )J

solves that equation as well. Since the solution is unique we get K q)(Cq∗ ) = (H ∗ H)−1 ,

K := JK (Cq∗ , en )−1 qn .

The left-hand product K q)(Cq∗ )

(5.2)

is known as Barnett’s factorization of Bd (q) [[14], p. 204], hence Bd−1 (q) = H ∗ H. Finally, Parseval’s identity [[29], p. 85] provides 3 2π ∗ it 1 Bd−1 (q) = H ∗ H = 2π ψn (e )ψn (eit )|q(eit )|−2 dt = BT−1 (q). 0 Suppose now Bd (q) = BT (q). Because BT (q) is positive, the assumption yields the positivity of Bd (q). With (3.2) relation σ(q) ⊂ D follows. As we have seen BT−1 (q) solves the Stein equation (5.1). To create for G and BT (q) corresponding equations, we generate their right-hand sides, respectively, by q with h deﬁned as in (2.1) and by the coeﬃcient vector v of the ﬁrst residue v of ) respect to q: v(z) = ψn (z)v,

v(z) := ) q(z) − a0 q(z),

a0 := q0 /qn .

Proposition 5.2. Supposed σ(q) ⊂ D the Gramian G and the Bezoutian BT (q) solve, respectively, Cq∗ XCq − X = −hh∗ ,

Cq XCq∗ − X = −vv∗ .

(5.3)

Proof. In view of (1.3), Proposition 5.1, R = p(Cq ) and e∗n R = qn h∗ with the abbreviation B := BT (q) we obtain Cq∗ GCq − G = R∗ (Cq∗ B −1 Cq − B −1 )R = −R∗ en |qn |−2 e∗n R = −hh∗ . According to the proof of Proposition 5.1 the matrix B can be replaced by K q)(Cq∗ ). Thus Cq BCq∗ − B = Cq K q)(Cq∗ )Cq∗ − K q)(Cq∗ ) = (Cq KCq∗ − K)) q(Cq∗ ). q(Cq )e1 . The identity Cq KCq∗ − K = −ve∗1 comThe deﬁnition of v implies v = ) pletes the proof.

Bezoutians Applied to Least Squares Approximation

267

Let Q ∈ Cn×n be the observability Gramian of Σ := (A, B, C) with respect to γ meaning that Q solves the matrix equation Lγ (A∗ , X) = −ΔC ∗ C,

Lγ (A, X) := γ00 X + γ01 AX + γ10 XA∗ + γ11 AXA∗ .

Here, Σ is a minimal realization of h meaning that the size of A is small as possible. For stable h the matrix Q exists and is unique. Consideration of the controllability realization Σco of h reveals G as the observability Gramian of Σco with respect to the unit circle. For σ(q) ⊂ D Proposition 5.3 states the equivalence of deﬁnition (2.3) of Q, deﬁnition (10) of Qn in [12] and deﬁnition (3.4) of χLS in [21]. As usual, let the Moore-Penrose pseudo inverse of A be denoted by A† [[18], p. 243]. † Proposition 5.3. Supposed σ(q) ⊂ D the minimizer ξ of F satisﬁes ξ = Hm V where ∞ −(k+1) Hm := [hi+j ]∞,m−1 V := col(hi )∞ := p(z) (5.4) i=m , i,j=0 , k=0 hk z q(z) . ∗ ∗ Hm and gm = Hm V . Thus, Proof. Parseval’s identity states Gm = Hm † ∗ ∗ Hm V = (Hm Hm )−1 Hm V = ξ.

Hence, the vector Hm ξ represents the projection of V on the linear subspace Hm := span(Hm ) := {Hm x : x ∈ Cm } ⊂ 2 meaning that the statement of Proposition 5.3 can be interpreted as projection method, moreover, since Hm possesses Hankel structure, as a semi-inﬁnite structured least squares problem. Together with formula (2.2) where x is to replace by ξ, the computation receipt proposed as in [12] is obtained. The projection point of view allows us to express the approximation error via the 2 -distance between V and its projection on Hm . Proposition 5.4. Let for σ(q) ⊂ D and x ∈ Cm the 2 -sequence ε be deﬁned by the components of the inﬁnite-dimensional vector Hm x − V . Supposed σ(Q) ∩ T = ∅ we have h − hx T2 ≤ ε2 / minT (Q). Proof. In virtue of Proposition 2.1 it suﬃces F (x) = ε2 to show. Obviously, the j th column of [Hm , V ] coincide with the Laurent coeﬃcient vector generated by fj . Thus, the deﬁnition of ε and the equation fk (z) = z k h(z) − ak (z) lead to m−1 ∞ E(z) := i=0 εi z −(i+1) = k=0 fk (z)xk − fm (z) = P (z) − h(z)Q(z). Together with px = Qp − P q we conclude E/Q = hx − h = −px /(Qq) ⇒ E = −px /q ⇒ ET2 = px Tdμ = F (x). Parseval’s identity yields ET2 = ε2 .

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S. Feldmann

6. The imaginary axis case γ = iR iR

The entries of G read as gij = ri |rj dμ where 3 iR 1 f (it)g(it)|q(it)|−2 dt. f |gdμ = 2π R To represent G as for the unit circle case the Hankel Bezoutian Bc (q) is in question. The name Hankel Bezoutian is justiﬁed by the fact that Bc−1 (q)I± is a Hankel matrix. Proposition 6.1. σ(q) ⊂ H− ⇔ Bc (q) = BiR (q). Proof. Using the bilinear transformation φ(z) := z+1 z−1 : D → H− , the generating functions (3.3) imply Bd (qφ ) = 2Mφ Bc (q)Mφ∗ . The φ-transformation of q and rj leads to ; ;
j

|qφ |2

row(rjφ )n−1 j=0

and Proposition 5.1 the equations Thus, with rj (z) = z , Rφ := n−1 n−1

= =

2π|qφ |2

i,j=0

i,j=0

2Rφ∗ BT−1 (qφ )Rφ = 2Rφ∗ Bd−1 (qφ )Rφ Rφ∗ Mφ−∗ Bc−1 (q)Mφ−1 Rφ = R∗ Bc−1 (q)R

are obtained. Finally, R = I completes the proof of “⇒”. Supposed Bc (q) = BiR (q) the Bezoutian Bc (q) is positive such that (3.2) implies σ(q) ⊂ H− . As we have just stated, for stable q the (i, j)-entry of Bd−1 (q) and Bc−1 (q) admits the representation 3 i j 1 z z |q(z)|−2 ds. 2π γ Supposing σ(q) ⊂ {z ∈ C : 0 < !(z)} such a representation 3 i+j 1 t |q(t)|−2 dt 2πi R of the (i, j)-entry of B(q, q)−1 have been proved already in [[22], Corollary 6.1]. Here, !(z) designates the imaginary part of the complex number z. To relate G and BiR (q) to Lyapunov equations, we generate the corresponding right-hand sides by h and by the coeﬃcient vector v of the ﬁrst residue v of q with respect to q: v(z) = ψn (z)v,

v(z) := q(z) − a0 q(z),

a0 := (−1)n qn /qn.

Note, that v coincides with the last column of BiR (q). Proposition 6.2. Supposed σ(q) ⊂ H− the Gramian G and the Bezoutian BiR (q) solve, respectively, Cq∗ X + XCq = −hh∗ ,

Cq X + XCq∗ = −vv∗ .

(6.2)

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269

Proof. Using the abbreviations B := BiR (q) and K := H (q1 , . . . , qn ) Barnett’s factorization of B reads as I± K ∗ q (Cq∗ ). The deﬁnition of v implies v = − q(Cq )e1 a0 . In addition one concludes K ∗ Cq∗ = Cq K ∗ ,

(Cq I± + I± Cq )qn = v(−1)n e∗n ,

Ken = e1 qn .

Using that identities one gets Cq B + BCq∗

= Cq I± K ∗ q (Cq∗ ) + I± K ∗ q (Cq∗ )Cq∗ = (Cq I± K ∗ + I± K ∗ Cq∗ ) q (Cq∗ ) ∗ ∗ −1 n ∗ ∗ ∗ = (Cq I± + I± Cq )K q (Cq ) = vqn (−1) en K q (Cq ) (−1)n qn e∗1 q (Cq∗ ) = −vv∗ . = v q−1 n = a0

With G = R∗ B −1 R, Cq R = RCq and e∗n R = qn h∗ one obtains Cq∗ G + GCq = R∗ (Cq∗ B −1 + B −1 Cq )R = R∗ B −1 (BCq∗ + Cq B)B −1 R −1 ∗ ∗ = −R∗ B −1 vv∗ B −1 R = −R∗ en q−1 n qn en R = −hh .

Note, that the solution property of Bc−1 (q) with respect to Cq∗ X + XCq = −en e∗n is a special case of [[23]. Theorem 4.1]. Consideration of the controllability realization Σco of h reveals G as the observability Gramian of Σco with respect to iR. As a consequence of Theorem 4.4 m for real h all coeﬃcients of Q are non-negative. Thus, the relation G−1 m g m ∈ R− can be stated where R− := {r ∈ R : r ≤ 0}. As in the unit circle case, ξ can be obtained as solution to a least squares problem in 2 . † Proposition 6.3. Supposed σ(q) ⊂ H− the minimizer ξ of F satisﬁes ξ = Mm Vm where ∞ r (z) −(i+1) Vm := col(him )∞ := qjφ . Mm := [hij ]∞,m−1 i=0 , i,j=0 , i=0 hij z φ (z)

Proof. Parseval’s identity states Bd−1 (qφ ) = H ∗ H where H is generated by 1/qφ . From above we know that G = 2Rφ∗ Bd−1 (qφ )Rφ . Thus Mn = HRφ , 2Mn∗ Mn = G and † ∗ ∗ Mm Vm = (Mm Mm )−1 Mm Vm = G−1 m gm = ξ. As counter part of Proposition 5.4 we have Proposition 6.4. Proposition 6.4. Let for σ(q) ⊂ H− and x ∈ Cm the 2 -sequence ε be deﬁned by the components of the inﬁnite-dimensional vector Mm x − Vm . Supposed σ(Q) ∩ iR = ∅ √ 2ε2 / miniR (Q). we have h − hx iR 2 ≤

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S. Feldmann

Proof. According to Proposition 2.1 it suﬃces to show F (x) = pxφ (z) qφ (z)

=

√

2ε2 . Obviously,

m−1 rmφ (z) rjφ (z) − xj qφ (z) qφ (z) j=0

=

∞

him z −(i+1) −

i=0

m−1 j=0

xj

∞

hij z −(i+1) =

i=0

∞

εi z −(i+1) .

i=0

Thus, transformation formula (6.1) and Parseval’s identity lead to 3 11 px (it) 112 3 2π 11 p (eit ) 112 ∞ 1 2 F (x)2 = 2π dt = π1 0 1 qxφ it 1 dt = 2 i=0 |εi | . R 1 q(it) 1 φ (e )

For the line case only σ(hξ ) ⊂ C is stated. Numerical experiences collected as in Example 2 show that actually the relation σ(Q) ⊂ γ holds, supposed the degree m of Q is less than δ where δ := n − κ,

n := deg q,

κ := deg p,

h = p/q.

In addition, for m = δ the polynomial Q becomes stable. To suppress technical diﬃculties, we prove these observations only for real h and γ = iR. Moreover, for m ∈ {δ + 1, . . . , n − 1} and accidently generated p and stable q the polynomial Q becomes always stable. In [11] it is proved that this situation is generic. For acquainting the reader with the proof idea for the case m ≤ δ we consider the Lyapunov equations (6.2). As a consequence the entries gij of G satisfy gij + gi−1,j+1 = −hi−1 hj ,

i = 1, . . . , n,

j = 0, . . . , n − 1.

(6.3)

Thus, setting gi := gii and hij := hi hj for real h the coincidence of [G4 |g4 ] with ⎤ ⎡ h11 h00 −g1 − h01 − h02 g2 − h03 + h12 g0 − ⎥ ⎢ 2 2 ⎥ ⎢ h11 h22 ⎥ ⎢ − h00 g1 −g2 − h12 − h13 − ⎥ ⎢ 2 2 2 (6.4) ⎥ ⎢ h11 h22 ⎢ −g − h g2 − − −g3 − h23 ⎥ ⎥ ⎢ 1 01 2 2 ⎦ ⎣ h22 h33 h11 − h02 −g2 − h12 g3 − − 2 2 2 is obtained. Consequently, for h0 = h1 = h2 = 0 the product G4 I± becomes a Hankel matrix generated by the vector [g0 , 0, −g1 , 0, g2 , 0, −g3 ]T and g4 is equal to we set [g2 , 0, −g3 , β]T , β := −h33 /2. Here, depending on the vector col(hi )2m−2 i=0 H(h0 , . . . , h2m−2 ) := [hi+j ]m−1 i,j=0 . The zero distribution in the coeﬃcient matrix and the right-hand side of the equation system Gm x = gm can be used now to give a mathematical foundation for our observations. But at ﬁrst for m = δ we relate the Gramian Gδ to the Bezoutian Bc (Q).

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m be related by Gm ξ = gm (β) Proposition 6.5. Let ξ ∈ Rm and col(gi )m−1 i=0 ∈ R where

Gm := HI± , H := H(f0 , 0, f1 , 0, f2 , . . . , fm−2 , 0, fm−1 ), fi := (−1)i gi , β ∈ R and

⎧ ' (T m−1 ⎨ 0, (−1) 2 g m+1 , 0, . . . , 0, gm−2 , 0, −gm−1, β , if m is odd, 2 gm (β) := T m ⎩ (−1) 2 g m2 , 0, . . . , 0, gm−2 , 0, −gm−1 , β , if m is even.

Then for invertible Gm the polynomial Q(z) := z m − ψm (z)ξ satisﬁes Gm Bc (Q) = −2βI. Proof. According to [[24], p. 19, Theorem 1.1 ] we have T ψm (x)H −1 ψm (y)(x − y) = Tα (x)S(y) − S(x)Tα (y)

where the polynomials Tα and S are deﬁned by Tα (z) := z m − ψm (z)H −1 f (α), S(z) := ψm (z)H −1 em , = T 0, f(m+1)/2 , . . . , 0, fm−2 , 0, fm−1 , α , if m is odd, f (α) := T fm/2 , 0, . . . , 0, fm−2 , 0, fm−1 , α , if m is even. Here, α represents an arbitrary real number. The substitution x := −x and the equation Gm = HI± lead to T B(x, y) := ψm (x)G−1 m ψm (y)(x + y) = S(x)Tα (y) − Tα (x)S(y),

S(z) := S(−z).

For odd m we have I± gm (β) = f (β). Thus Q(z) = z m − ψm (z)H −1 f (β) = Tβ (z). Since the distribution of the vanishing entries in H will be preserved by its inverse, together with the zero distribution in f (β) it follows that the product (I± + I)H −1 (f (β) − em β) is equal to the zero vector. Thus Q(z) + Q(z) = Tβ (−z) + Tβ (z) = −ψm (z)(I± + I)H −1 f (β) = −ψm (z)(I± + I)H −1 (f (β) − em β + em β) = −2βψm (z)H −1 em = −2βS(z). − Q = −2βS. Thus On the same way for even m we get Q = T−β and Q −2βB(x, y)

= −2β S(x)T ±β (y) − T±β (x)(−2βS(y)) = (Q(x) ∓ Q(x))Q(y) − Q(x)( Q(y) ∓ Q(y)) Q(y). = Q(x)Q(y) − Q(x)

Finally, in view of the generating function (3.3) the statement follows.

Proposition 6.6. Let h ∈ R(z) be stable. Then for m < δ we have σ(Q) ⊂ iR and for m = δ the polynomial Q becomes stable.

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Proof. Due to δ = deg q − deg p the Laurent coeﬃcients h0 , . . . , hδ−2 of h vanish. Hence, in view of (6.3) the entries of Gδ satisfy gij + gi−1,j+1 = 0,

i = 1, . . . , δ − 1,

j = 0, . . . , δ − 2.

(6.5)

That equation establishes a recurrence relation within the entries along every antidiagonal of Gδ . Here, the kth anti-diagonal Ak is formed by all entries gij for which i + j = k. For k = 2i − 1 equation (6.5) passes over to gi,i−1 + gi−1,i = 0. Since h is real, we have gij = gji . Thus, gi,i−1 = 0 holds. Consequently, all entries of A2i−1 vanish. For k = 2i equation (6.5) passes over to gii + gi−1,i+1 = 0. Thus, all entries of A2i are equal to gi or −gi . Collecting these observations we get Gδ = H(g0 , 0, −g1 , 0, . . . , 0, (−1)δ−1 gδ−1 )I± . For m < δ the structure of Gδ leads to a polynomial Q(z) := z m − ψm (z)G−1 m gm whose coeﬃcients of all even or of all odd powers vanish. In addition, Theorem 4.4 states σ(Q) ⊂ H− ∪ iR. Since all zeros of such a polynomial are located on iR, the ﬁrst statement is proved. For m = δ the vector ξ satisﬁes Gδ ξ = gδ (−h2δ−1 /2). Thus, ξ and col(gi )δ−1 i=0 are related as in Proposition 6.5. In combination with hδ−1 = pκ the matrix equation Gδ Bc (Q) = p2κ I is obtained. Since h is stable, the Gramian Gδ is positive. In combination with pκ = 0 the same holds for Bc (Q). Finally, (3.2) yields the stability of Q. As mentioned above, the statement of Proposition 6.6 for m < δ is a simple consequence of Fej´er’s convex hull theorem, the case m = δ have been treated here in detail and the case m ∈ {δ + 1, . . . , n − 1} is subject of [11].

7. Computation of hξ via its minimal realizations There is a third approach to get hξ . Depending on a minimal realization Σ := (A, B, C) of h we set K := K (A, B) and consider the observability Gramian Q of Σ with respect to γ. Proposition 7.1. For stable h the Gramians G and Q are related by G = K ∗ QK. Proof. The minimality of Σ provides the invertibility of K and the stability of h the existence of Q. Immediately by deﬁnition the equations CK = h∗ ,

AK = KCq

follow. It turns out that (5.3) and (6.2) can be generalized to Lγ (Cq∗ , G) = −Δhh∗ that means G represents the observability Gramian of the controllability realization Σco with respect to γ. Thus, we have Lγ (A∗ , K −∗ GK) = K −∗ Lγ (Cq∗ , G)K −1 = −ΔK −∗ hh∗ K −1 = −ΔC ∗ C. Finally, the uniqueness of Q provides the statement.

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n−1 n Assuming that A = diag(αi )n−1 i=0 , B = col(1)i=1 , and C = row(cj )j=0 , we deﬁne the Vandermonde matrix V and the Cauchy matrix Ω by

n−1 −Δci cj j n−1 V := [αi ]i,j=0 , Ω := . (7.1) pγ (αj , αi ) i,j=0

Then K = V and Q = Ω such that supposed αi ∈ C the Gramian G admits the factorization G = V ∗ ΩV . Supposed that all poles of h are simple, we call Ω the Cauchy matrix generated by h and γ. For numerical examples which are generated by prescribed αi ∈ C and ci ∈ C the matrix G can be obtained by the product V ∗ ΩV . In particular, for α, β ∈ C the identity 3 |w(t)|dt ˙ = p−2πΔ , I ∈ {[0, 2π], R}, γ ∈ {T, iR} I (w(t)−α)(w(t)−β) γ (β,α) holds where w(t) is deﬁned as in the proof of Proposition 4.3. With the abbreviation V := Km (A, B) we get V ∗ QV = Gm meaning that V ∗ QV is invertible and appropriate to deﬁne Z := (V ∗ QV )−1 V ∗ Q. Proposition 7.2. The triple (ZAV, ZB, CV ) realizes hξ . Proof. We have ZAV = (V ∗ QV )−1 V ∗ Q[AB, . . . , Am B] = [e2 , . . . , em , (V ∗ QV )−1 V ∗ QAm B]. Together with Gm = V ∗ QV and gm = V ∗ QAm B one gets ZAV = CQ . Finally, ZB = (V ∗ QV )−1 V ∗ QB = e1 ,

CV = [h0 , . . . , hm−1 ]

thus, above triple represents the controllability realization of hξ .

8. The construction of hηξ The renouncement on the disappearance of the ﬁrst m Laurent coeﬃcients of h−hξ admits to proceed with a second optimization step resulting in a rational function hηξ with h − hηξ γ2 ≈ h − hopt γ2 . Instead to apply (2.2), the numerator polynomial P of hxξ will be deﬁned now via the minimizer η ∈ Cm of m−1 1 k F : Cm → R, F (x) := hxξ − hγ2 , hxξ (z) := Q(z) k=0 xk z such that hηξ := P/Q, P(z) := ψm (z)η. As in Section 2 the minimizer ξ, the minimizer η turns out to be the solution to > i ?γ m−1 ?γ m−1

> i z zj z x = W, W := col . (8.1) h Q Q 2 i,j=0 Q 2 i=0 Therefore, for the determination of η it suﬃces to ﬁnd the stable polynomial Q and to apply (8.1). In addition, the deﬁnition of Bγ (Q) implies η = Bγ (Q)W . That

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approach becomes essential for the case where the ﬁrst m Laurent coeﬃcients of h are small in comparison with the subsequent coeﬃcients. Then according to (2.2) the numerator of hξ becomes approximately the zero polynomial, a circumstance which does not lead to satisfying approximation results. Here, we say that a complex number is small, if its magnitude is small. For h := 1/q, γ := T and m := n−1 the approximant hηξ can be expressed explicitly in terms of q: hηξ (z) =

|qn |2 − |q0 |2 + q(z)q0 − ) q(z)qn . ) qn (q(z)qn − q (z)q0 )

Note, that z represents a common divisor of the numerator and the denominator such that the degree of hηξ is actually equal to n − 1. Moreover, that formula reveals an interesting interpolation property of hηξ . Let β be a pole of hηξ . Then its q (β)q0 . Supposed β = 0 together denominator vanishes in β such that q(β)qn = ) with q)(z) = z n q(1/z) that equation implies ) n −) ) 0 ) = q(β)(|q ) n |2 − |q0 |2 ), q(β)q ) 0 −) ) n = 0, β) := β −1 . qn (q(β)q q(β)q q(β)q ) = q(β) ) −1 = h(β) ) holds. The coincidence of hη and h on the mirror Thus, hηξ (β) ξ image set φγ (σ(hηξ )) along γ will be maintained, if one sets φγ (z) := −

γ00 + γ01 z . γ10 + γ11 z

Proposition 8.1. Let h be stable and β ∈ σ(hηξ ). Then h(φγ (β)) = hηξ (φγ (β)). Proof. Since h and hηξ are strictly proper for inﬁnite φγ (β) the statement holds. Suppose |φγ (β)| < ∞. To avoid technical details, we prove the implication only for m real h and γ = T meaning that φγ (β) = β −1 . Let Q(z) := k=0 Qk z k be an arbitrary real stable polynomial of degree m. Combination of Barnett’s factorization (5.2) with JBd (Q)J = Bd (Q) leads to ) Q )H , Bd (Q) = J Q(C

H := H (Q1 , . . . , Qm ).

i e 1 )∞ The Hankel matrix Θ generated by 1/Q fulﬁlls Θ∗ = H−1 row(CQ i=0 , thus the vector W deﬁned as in (8.1) satisﬁes ∞ −1 i ∞ ∞ −1 i W = Θ∗ col(hi )∞ i=0 = H row(CQ e1 )i=0 col(hi )i=0 = H i=0 CQ e1 hi .

First, we consider for k > 0 the special case h(z) := z −k . Then all Laurent coeﬃcients hi of h(z) vanish excepted hk−1 which is equal to 1. Consequently, W simpliﬁes to k−1 W = H−1 CQ e1 such that the coeﬃcient vector ηk of the numerator polynomial Pk of hηξ k generated by Q and z −k admits the representation ) Q )C k−1 e1 . ηk = Bd (Q)W = J Q(C Q

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k = β k ψm (β). Thus, The assumption Q(β) = 0 implies now ψm (β)CQ −1 ) Q )C k−1 e1 Pk β = β 1−m ψm (β)Jηk = β 1−m ψm (β)Q(C Q k−1 = βQ(β −1 )ψm (β)CQ e1 = β k Q(β −1 ) ∞ proving the statement for h(z) := z −k . Finally, let h(z) := k=0 hk /z k+1 be real and stable. Then the coeﬃcient vector η of P admits the representation ∞ η = k=1 ηk hk−1

such that h(β −1 )Q(β −1 ) =

∞ k=1

hk−1 β k Q(β −1 ) =

∞ k=1

Pk (β −1 )hk−1 = P(β −1 ).

Because h, Q, P are real the equation h(β −1 )Q(β −1 ) = P(β −1 ) holds.

In the case where all zeros of Q are simple, the computation of η via Bγ (Q)W can be replaced by solving the interpolation problem P(γi ) = h(γi )Q(γi ),

γi := φγ (βi ),

i = 0, . . . , m − 1.

Example 1 and 4 show the improved performance of

hηξ

(8.2)

in comparison with hξ .

9. The limit behavior of Q For simpliﬁcation let γ ∈ {T, iR}, h ∈ R(s) and q(z) = the limit behavior of Q, remember its sum representation n Q(z) = j=m+1 ζj mj (z)

n−1 i=0 (z

− αi ). To study

provided as in Proposition 4.1, the equation R = p(Cq ) and the deﬁnition of the d q polynomial v and the vector w as in the end of Section 3. As usual we set q := dz and assume the stability of h. Vanishing q(α) provides ψn (α)p(Cq ) = p(α)ψn (α). Therefore, w(z) := ψn (z)w satisﬁes the interpolation condition w(α) = v(α)/p(α). We establish now a recurrence relation within the numbers mj (α) and express mn (α) explicitly. With the deﬁnition qφγ (z) := q(φγ (z))(γ10 + γ11 z)n we get qφ = q) and qφ T

iR

= q . In addition, we set K(x, y) := −

r(x)r(y) , pγ (x, y)

r :=

q φγ qp

.

Proposition 9.1. For α ∈ σ(q) the numbers (mj (α))nj=0 satisfy the recurrence relation γ = T : mj+1 (α) = αmj (α) − αqj mn (α) + wj q)(α)/p(α) (9.1) γ = iR : mj (α) = mn (α)qj − αmj+1 (α) + (−1)n+1 wj q (α)/p(α) where m0 (z) ≡ 0. In the case where all zeros αi of q are simple, the polynomial mn (z) satisﬁes the interpolation conditions n−1 (9.2) mn (αi ) = q (αi ) j=0 K(αi , αj ), i = 0, . . . , n − 1.

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Proof. In the circle case the equation Cq M Cq∗ − M = −ww∗ holds. Thus the companion structure of Cq provides for the columns of M the equation mj+1 = Cq (mj − mn qj ) + wwj . Left multiplication with ψn (α) yields mj+1 (α) = α(mj (α) − mn (α)qj ) + w(α)wj . Since w(α) = v(α)/p(α) and v(α) = q)(α) we have w(α) = q)(α)/p(α). In the line case M satisﬁes Cq M + M Cq∗ = −ww∗ . Thus the companion structure of Cq provides for the columns of M the equation mj = mn qj − Cq mj+1 − wwj . Left multiplication with ψn (α) yields mj (α) = mn (α)qj − αmj+1 (α) − w(α)wj . q (α) Since w(α) = v(α)/p(α) and v(α) = (−1)n+1 q (α) we have w(α) = (−1)n+1 p(α) . To prove in the circle case the validity of the asserted interpolation conditions, we introduce the diagonal matrix

D := diag () q (αi ))n−1 i=0 . Then the generating function of Bd (q) yields V Bd (q)V ∗ = DΩD ∗ ,

V := [αji ]n−1 i,j=0 ,

Ω := [(1 − αi αj )−1 ]n−1 i,j=0 .

Furthermore, R admits the factorization V −1 diag(p(αi ))n−1 i=0 V such that

n−1 q)(αi ) −1 −1 −∗ −1 ∗ −∗ M = G = R Bd (q)R = V DΩD V , D := diag . (9.3) p(αi ) i=0 ∗ Since the last column of V −∗ is equal to col(q (αi )−1 )n−1 i=0 and ψn (αi ) = ei V we have mn (αi ) = ψn (αi )M en = e∗i DΩD ∗ col(q (αi )−1 )n−1 i=0

n−1 n−1 q)(αi ) ∗ q)(αj ) q)(αi ) q)(αj ) = = ei Ωcol . p(αi ) p(αj )q (αj ) j=0 p(αi ) j=0 (1 − αi αj )p(αj )q (αj )

The proof of the statement with respect to iR is left to the reader.

We describe now the inﬂuence of p on σ(hξ ). The statements are formulated and proved only for the unit circle, but to ensure their validity for the half-plane Example 3 concerns that case. Instead of considering the variation of p, we consider the variation of ϕ := p(α) ∈ C, α ∈ σ(q). Since the associated partial fraction coeﬃcient reads as ϕ/q (α), the case where that coeﬃcient tends to inﬁnity is in question. The dependency of a function f and a matrix M on ϕ will be designated by f (., ϕ) and Mϕ , respectively. In addition, we use the abbreviation M (k, ) := [mij ] i,j=k . For simpliﬁcation of the proofs we assume αi = αj . Proposition 9.2. Let α ∈ σ(q) ⊂ D. Then α is a zero of limϕ→∞ Q(., ϕ).

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Proof. W.l.o.g. let α := α0 . It suﬃces to show the existence of the limit vector limϕ→∞ ζϕ arising in the sum representation of Q(., ϕ) and that α becomes a zero of all polynomials mj (., ϕ) as ϕ → ∞. According to Proposition 9.1 we have lim mn (α, ϕ) = q (α)

n−1

lim K(α, αj , ϕ) = 0. =0 The recurrence relation (9.1) implies limϕ→∞ mj (α, ϕ) = 0. In factorization (9.3) only D depends on ϕ. Therefore, m0 (α, ϕ) = 0,

ϕ→∞

j=0

ϕ→∞

∗ −∗ Mϕ = V −1 Dϕ ΩDϕ V .

Since αi = αj and σ(q) ⊂ D the Vandermonde matrix V is regular and the Cauchy matrix Ω is positive. Let c1 col(zi )ni=1 := row(cj )nj=1 := V −1 , z∗i ∈ Cn−1 , cj ∈ Cn . Since the (1, 1)-entry of Dϕ tends to zero as ϕ → ∞ the limit M∞ := limϕ→∞ Mϕ satisﬁes ∗ M∞ = col(zi )ni=1 D(2, n)Ω(2, n)D(2, n) row(z∗i )ni=1 . Since e∗1 V e1 = 1 Cramer’s rule implies the independence of {z∗2 , . . . , z∗n }. Thus, for k > 1 the sub-matrices ∗

M∞ (k, n) = col(zi )ni=k D(2, n)Ω(2, n)D(2, n) row(z∗i )ni=k are invertible. Finally, limϕ→∞ ζϕ = M∞ (m + 1, n)−1 e1 .

We observe now the behavior of σ(Q(., α)) if the conjugate complex pair {α, α} ⊂ σ(q) ⊂ D tends to the dotted boundary Tp := T \ {σ(p) ∪ {±1}}. Proposition 9.3. For m > 1 and α ∈ σ(q) ⊂ D the relation α ∈ limα→Tp σ(Q(., α)) holds. Proof. It suﬃces to show that limα→Tp mj (α) = 0 and that for k > 2 the limit limα→Tp Mα (k, n) is invertible. W.l.o.g. let {α0 , α1 } = {α, α}. According to Proposition 9.1 we have n−1 limα0 →Tp mn (α0 ) = q (α0 ) j=0 limα0 →Tp K(α0 , αj ) K(α0 , αj ) =

n−1 i=0,i =j (1 − α0 αi ) . q (α0 )p(α0 )q (αj )p(αj )

q)(αj )

)(αj ), Because q)(z) = n−1 i=0 (1 − zαi ) and α1 = α0 , for j = 0 the second factor of q n−1 and for j = 0 the ﬁrst factor of i=0,i =j (1 − α0 αi ) is equal to 1 − |α0 |2 . Thus K(α0 , αj ) → 0 as α0 → Tp . Regarding (9.1) by induction limα→Tp mj (α) = 0 follows. To show the invertibility of limα→Tp Mα (k, n) observe that all factors in (9.3) depend on α: Mα = Vα−1 Dα Ωα Dα∗ Vα−∗ .

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Obviously, for α → Tp the both ﬁrst main diagonal entries of Dα vanish. The (1, 1)- and the (2, 2)-entry of Ωα are equal to 1/(1 − |α|2). Thus, the corresponding entries of Dα Ωα Dα∗ coincide with q)(α)) q (α) . p(α)(1 − |α|2 )p(α) Since q)(α)) q (α) possesses the factor (1 − |α|2 )2 the limit limα→T Consequently, MT := limα→Tp Mα satisﬁes

q)(α)) q (α) 1−|α|2

vanishes.

n −1 MT = row(cj )nj=3 D(3, n)Ω(3, n)D(3, n)∗ row(cj )n∗ . j=3 , row(cj )j=1 := ( lim Vα ) α→Tp

Note that ±1 ∈ Tp ensures the invertibility of limα→Tp Vα . Finally, the invertibility of the main sections MT (k, n), k > 2, is obtained as in the proof of Proposition 9.2. For γ = iR Example 3 illustrates the validity of Propositions 9.2 and 9.3.

10. Bilinear transformation between D and H− The least squares method for the discrete case can be applied to the continues case as follows. Let φ(z) := z−1 z+1 and hφ be deﬁned by hφ (z) := h(φ(z)). For 1 ∈ σ(A) the identity (φ(z)I − A)−1 = (I − A)−1 + 2(I − A)−2 (zI − (I + A)(I − A)−1 )−1 holds. Due to σ(Ac ) ⊂ H− we have 1 ∈ σ(Ac ). Thus, supposed (Ac , Bc , Cc ) realizes h, the rational function hφ admits the representation hφ (z) = Cc (I − Ac )−1 Bc + 2Cc (I − Ac )−2 (zI − (I + Ac )(I − Ac )−1 )−1 Bc (10.1) =: Cd =: Ad = h(1) meaning that (Ad , Bc , Cd ) realizes hφ −h(1). Consequently, the Laurent coeﬃcients hφk of hφ read as (Cd Akd Bc )∞ k=0 , the vector ωφ can be computed according to (2.2) and due to σ(Ad ) ⊂ D the vector ξφ can be computed as the vector ξ in Proposition 5.3. Finally, as approximant of h the rational function

Pφ 1+z moeb , hξφ := h(1) + hred (z) := hξφ 1−z Qφ is in question. Example 5 compares the approximant hξ obtained directly with the approximant obtained as just carried out.

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11. Examples m According to (2.3) for γ ∈ {T, iR} the minimizer ξ := col(ξi )m−1 of F can i=0 ∈ C m−1 be computed as solution to Gm x = gm where Gm := [gij ]i,j=0 , gm := col(gim )m−1 i=0 and 3 2π 1 fi (eit )fj (eit )dt γ = T : gij := 2π 0 3 +N 1 γ = iR : gij := limN →∞ 2π f (it)fj (it)dt (11.1) −N i

fk (z) := CAk (zI − A)−1 B. By approximation of h via hξ the coeﬃcient vector ω of its numerator polynomial P has to be computed according to (2.2) where x has to be replaced by ξ. By approximation of h via hηξ the coeﬃcient vector η of the numerator polynomial P of hηξ can be obtained by Bγ (Q)W where W is deﬁned as in (8.1), or equivalently as solution to the interpolation problem (8.2) supposed that the zeros βi of Q are known and simple. Thus, ⎡ ⎤ h0 . . . hm−1

−col(ξi )m−1 ⎢ ⎥ . . i=1 .. .. ω := ⎣ ⎦ 1 h0 P > i ?γ m−1 hξ := , P := ψm ω, z Q η := Bγ (Q)W, W := col h Q := z m − ψm ξ, Q 2 i=0 P η , P := ψm η, hξ := or equivalently Q −1 m−1 col (h(γi )Q(γi ))i=0 η := [γij ]m−1 i,j=0 γi := φγ (βi ). (11.2) Examples 1, 2, 5 use that computation receipt. If one refuses the computation of the ∞ −(k+1) integrals (11.1) and if large enough subsections of (hk )∞ , h(z) = h z , k=0 k=0 k ∞ ∞ −1 −(k+1) and (θk )k=0 , Q(z) = k=0 θk z , are known, then in the unit circle case the vectors ξ and η can be obtained as limits of the truncated computation procedure †

N,m−1 +m N N col(hi )N ξN := Hm i=m , Hm := [hi+j ]i,j=0 , limN →∞ ξN → ξ †

N ηN := ΘN m col(hi )i=0 ,

N,m−1 ΘN m := [θi+j ]i,j=0 ,

limN →∞ ηN → η

(11.3)

which has been used in Example 4. If one knows q and the residues rj , and is able to compute qφ and rjφ , then according to Proposition 6.3 for the continues case the minimizer ξ can also be obtained via a projection method applied to elements of 2 . But when σ(h) is close to γ, only the application of solution algorithms, N which address the Hankel structure of Hm and ΘN m , lead to satisfying numerical results. Thus, it is natural to ask for algorithms which compute H † by exploitation of its structure. In the literature the Toeplitz structure is exploited in two diﬀerent directions: one approach leads to algorithms of low complexity (fast and super

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fast algorithms) and a second approach to algorithms which take care for accuracy and stability (high performance algorithms). For fast and super fast algorithms consult among others [1, 2, 3, 6, 7] and for high performance algorithms consult [16, 17, 20]. The paper [19] goes in both directions. Numerical problems arising in connection with the computation of the integrals (11.1), of the sequences (hk )∞ k=0 N† N† and (θk )∞ , or of H and Θ are not addressed in this paper. Supposed a m m k=0 fraction representation p/q of h is known and γ ∈ {T, iR}, the Bezoutian Bγ (q) can be computed via (3.1) and G via the product p(Cq )∗ Bγ−1 (q)p(Cq ). In Example 3 and 6 we use that formula, but for large n the inversion of Bγ (q) becomes very expensive which reﬂects the disadvantage of that approach. Finally, in the case where a minimal realization of h is available Proposition 7.2 can be applied. A disadvantage of that computation approach is the necessity to know the corresponding observability Gramian Q completely. To overcome associated storage and condition problems, in [28] a low rank approximation of Q is recommended which is found by a low-rank Smith type method. Observe that by exploitation of Gm x = gm the knowledge only of the m × (m + 1)-dimensional left upper corner of G is necessary, and that according to (6.4) the matrix [Gm , gm ] is completely determined by the numbers gii , hi , i = 0, . . . , m − 1. The approximation procedures will be illustrated now by means of some examples. Here, we pay our main attention to the line case, the special case h = 1/q and the situation where σ(h) is close to γ or the partial fraction coeﬃcient of a pole is large in comparison with the coeﬃcients of the other poles. For the unit circle and general p a lot of examples are available in [12, 13, 21]. Example 1. We consider the m-degree approximation of

κ κ 1 1 ck ck k 2 i − ∈ H h(z) := + , αk := , ck := k , c := ck − 2c z − αk z − αk 8 2κ k=1 k=1 κ with respect to γ := iR. Due to hk = c−1 i=1 ci αki for κ = 10 we have h0 h1 h2 h3 = 1 −0.049 −1.026 0.1680 . For N = 5000 the computation of the truncated integrals (11.1) yields ⎤ ⎡ 1.550 −0.500 −1.671 1.027 ⎢ −0.500 1.720 −0.001 −2.113 ⎥ ⎥ G4 = [gij ]3i,j=0 = ⎢ ⎣ −1.671 −0.001 2.063 −0.526 ⎦ 1.027 −2.113 −0.526 2.807 T T g4 = g04 . . . g34 = 1.945 0.534 −2.635 −0.014 what can be veriﬁed by utilization of (6.4) or of G = V ∗ ΩV where V and Ω are deﬁned as in (7.1). The solution ξ to G4 x = g4 and the vector ω computed via (11.2) read as T T ξ := − 0.865 0.652 2.126 0.578 , ω := 0.123 1.072 0.529 1 .

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2.5 "F.dat" "Fred.dat" 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 0

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Figure 1. n = 2κ = 20, m = 4, solid line = ) h(iR), dashed line = ) hξ (iR) 0.6 "h.dat" "h_xi.dat" "h_xi^eta.dat" 0.4

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Figure 2. n = 2κ = 200, m = 14, solid line = ) h(iR), dashed line = ) hξ (iR), dashed line = ) hηξ (iR) In Figure 1 the Nyquist plots of h and hξ are compared to each other. According to Theorem 4.4 the zeros of Q belong to H− . Actually, σ(Q) = {−0.18 ± 0.751i, −0.109 ± 1.2i} =: {β0 , . . . , β3 } holds. With φγ (z) = −z the solution to the interpolation problem (8.2) reads as T . η := 0.106 1.012 0.515 0.949 Since η ≈ ω there is no signiﬁcant diﬀerence between hξ (iR) and hηξ (iR). Enlargement of κ and m shows the improved performance of hηξ more better. Figure 2 plots for κ = 100 and m = 14 the sets h(iR), hξ (iR), hηξ (iR) where G14 and g14 have been generated by the factorization G = V ∗ ΩV . In particular h − hξ iR 2 = 0.093,

h − hηξ iR 2 = 0.087,

hiR 2 = 0.463.

Example 2. To illustrate the situation σ(Q) ⊂ γ we consider the line γ := {w(t) : t ∈ R}, w(t) := 1 + (2 + i)t

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and the monic polynomial q generated by the zeros α0 := −1 + 0.5i,

α1 := −0.1 + 1i,

α2 := 0.3 + 1.5i,

α3 := 1.5 + 1.1i.

It turns out, that σ(q) is located on the left-hand side of γ, if t runs over (−∞, +∞). The setting h := 1/q simpliﬁes the integral (11.1) to 3N i 1 gij = limN →∞ 2π ˙ dt. w(t) w(t)j |q(w(t))|−2 |w(t)| −N For N = 400 and m = 3 we get T . ξ = −0.972 − 1.819i −0.32 + 2.503i 2.358 − 0.321i Actually, the parameters t1 := −0.8325,

t2 := −0.0684,

t3 := 0.5795

satisfy σ(Q) = {w(t1 ), w(t2 ), w(t3 )}. Example 3. We give an example for the limit situations as described in Proposition 9.2 and 9.3. We have to consider a rational function h with the property, that some of its partial fraction coeﬃcients are large in comparison with the others. Let σ(q) be given by α0,1 := −1 ± i, α2,3 := −2 ± i and p be deﬁned via the interpolation conditions p(α0 ) = p(α1 ) = 50,

p(α2 ) = p(α3 ) = 1.

Then h := p/q where p(z) = −19.6z 3 − 88.2z 2 − 137.2z − 48,

q(z) = z 4 + 6z 3 + 15z 2 + 18z + 10.

For γ := iR and m := 2 the relation σ(Q) ≈ {α0 , α1 } is to be expected. To be exact we use G = p(CqT )Bc−1 (q)p(Cq ). Then we have

108.237 −192.08 166.685 1.984 −1 , g2 = . = G2 g2 , G2 = ξ=− −192.08 409.555 −432.18 1.986 Thus, actually σ(Q) = {−0.993 ± 0.999i} holds. To illustrate Proposition 9.3 we move α0,1 close to γ: α0,1 := −0.01 ± i, let α2 , α3 be unchanged and choose p according to p(α0 ) = p(α1 ) = 1, p(α2 ) = p(α3 ) = −1. As above ξ = −[0.99, 0.075]T is obtained, thus σ(Q) = {−0.037 ± 0.994i} ≈ {α0 , α1 } as expected. It turns out that for N = 103 the utilization of the truncated integrals (11.1) yields the same result. Example 4. To illustrate for n := 20 and large p0 ∈ R the advantage of a second optimization step as proposed in Section 8, we approximate h := p/q along T by rational functions hξ and hηξ of degree m := 7 where p(z) :=

n−1 k=1

n 2 −1

k

z + p0 , q(z) :=

k=0

(z − αk )(z − αk ), αk :=

3k 8k2 8 − − i. 10 n + 1 (n + 10)2

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1 cos(t), sin(t) "ZEROS.dat" "zerosxi.dat"

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8000 6000 4000 2000 0 -2000 -4000 -6000 -8000 -10000 -6000

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Figure 4. n = 20, m = 7, + := h(U), × := hηξ (U), ∗ := hξ (U) Figure 3 shows the distribution of the αk marked by “+”. We choose p0 := 10, η +m N N := 200, and form the sequences (hk )N k=0 and (θk )k=0 to ﬁnd hξ and hξ . Then (11.2) and (11.3) provide T

ξ = [0.336, −1.562, 2.889, −2.948, 2.879, −3.652, 3.057] , T ω = [0.677, 0.752, 0.769, 0.866, 0.786, 1.085, 1.000] , T η = [120.391, −226.509, 170.689, −143.321, 183.663, −126.761, 34.052] . For suﬃciently large N Theorem 4.4 states the relation σ(Q) ⊂ D. Figure 3 conﬁrms that localization where {β0 , . . . , β6 } := σ(Q) is marked with “×”. It is easy to check, that P, Q, h satisfy the interpolation conditions P(γi ) = h(γi )Q(γi ), γi := 1/βi . In Figure 4 the Nyquist plots of h, hξ , hηξ along the unit roots U := σ(z 3000 − 1) are compared to each other. Obviously, hηξ ﬁts more better than hξ , precisely @ @ 2 |h(w)−hη |h(w)−hξ (w)|2 w∈U ξ (w)| w∈U = 0.571, = 0.183. |h(w)|2 |h(w)|2 w∈U

w∈U

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Example 5. We approximate h(z) :=

2 k=0

1 1 + , α0 := −2 + 7i, α1 := −2 + 2i, α2 := −2 + 1i z − αk z − αk

of degree m := 3, and compare the corresponding along γ := iR by hξ and hmoeb red approximation quality. Obviously, h admits the fraction representation h(z) = Since hk := 2

6z 5 + 60z 4 + 456z 3 + 1776z 2 + 3570z + 2916 . z 6 + 12z 5 + 114z 4 + 592z 3 + 1785z 2 + 2916z + 2120

2

i=0

αki the ﬁrst 3 Laurent coeﬃcients of h read as h0 h1 h2 = 6 −12 −84 .

The computation of gii , i = 0, 1, 2, via the integrals (11.1) and application of (6.4) yield the system ⎤ ⎤ ⎡ ⎡ 576 4.832 −18 −8.934 ⎣ −18 80.934 −72 ⎦ x = ⎣ −2303.242 ⎦ . −3528 −8.934 −72 1295.242 Its solution ξ and the resulting ω read as T , ξ = − 152.464 69.145 7.619

ω :=

239.439 33.714 6

T

.

The Laurent coeﬃcients hφk of hφ are obtained via Cd Akd Bc where (Ad , Bc , Cd ) is deﬁned as in (10.1) and Ac := diag(α0 , α1 , α2 , α0 , α1 , α2 ) ∈ C6×6 , BcT := Cc := 1 . . . 1 ∈ R6 . For N = 190 the truncated computation procedure (11.3) provides as approximation of hφ along T the rational function hξφ := h(1) + Pφ /Qφ where T † φ 193 = [hφi+j ]190,2 Qφ (z) := z 3 −ψ3 (z)ξφ , ξφ := − 0.303 1.238 1.904 i,j=0 col(hi )i=3 ⎤⎡ ⎤ ⎡ φ ⎡ ⎤ h0 hφ1 hφ2 −ξ1φ 0.099 Pφ := ψ3 ωφ , ωφ := ⎣ 0.448 ⎦ = ⎣ hφ0 hφ1 ⎦ ⎣ −ξ2φ ⎦ h(1) = 1.165 0.391 1 hφ0 that means 1.165z 3 + 2.609z 2 + 1.889z + 0.453 Pφ (z) = . hξφ (z) = h(1) + Qφ (z) z 3 + 1.904z 2 + 1.238z + 0.303 Finally, hmoeb red (z) := hξφ

1+z 1−z

=

−0.243z 3 + 11.764z 2 + 94.772z + 202.879 . z 3 + 25.496z 2 + 91.439z + 147.452

with respect to iR are compared In Figure 5 the Nyquist plots of h, hξ and hmoeb red to each other. Obviously, for some intervals of iR the approximation quality diﬀers signiﬁcantly. Along i[−5, 5] the approximant hmoeb ﬁts better than hξ , and along red i(R \ [−5, 5]) the approximant hξ is to be preferred. Remember, a positive (real)

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Figure 5. a= ) h(iR), b= ) hξ (iR), c= ) hmoeb red (iR) function h(z) is a (real) function whose real part is positive when the real part of z is positive [[8], p. 197]. According to [[8], Theorem V] a real rational function h is positive, if σ(h) ⊂ H− and (h(iR)) ⊂ R+ . The deﬁnition of the numbers αi and Theorem 4.4 ensures the stability of h and hξ . Thus, Figure 5 shows the positivity of h and hξ , and the non-positivity of hmoeb red . Consequently, if one like to preserve positivity, then hξ is to be preferred. Unfortunately, also the direct approach does not preserve positivity in general. We ﬁnish the paper with a counter example. 2 Example 6. Let h(z) := k=0 ck (z − αk )−1 where α0 := −1 + i, α1 := α0 , α2 = −3, c0 := c1 := 1, c2 := 2. Then with p(z) := 4z 2 + 12z + 10 and q(z) := z 3 + 5z 2 + 8z + 6 the equation h = p/q follows. Because all ck are positive, and h is stable with respect to iR, we are dealing with a positive real function. To avoid the approximation of the integrals (11.1) we use factorization (1.3). The representation (3.1) of Bc (q) and R = p(Cq ) provide ⎤ ⎤ ⎡ ⎡ 10 −24 48 680 0 −816 816 0 ⎦ , R = ⎣ 12 −12 40 ⎦ Bc−1 (q) = |Bc1(q)| ⎣ 0 4 −8 18 −816 0 6528 |Bc (q)| = 55488. −1 (q)R∗ one gets the equation By exploitation of Bc (q) = BiR (q) and G = RBiR system

224672 −443904 884544 x= −443904 891072 −1775616 = G2 |Bc (q)| = g2 |Bc (q)|

to determine ξ ∈ R2 . Thus, ξ = −[0.0034, 1.9944]T. Since [h0 , h1 , h2 ] = eT 3 R we have h0 = 4 and h1 = −8 such that in view of (11.2) the coeﬃcient vector ω of the numerator polynomial of hξ becomes [−0.023, 4]. Consequently, hξ (0) = −ω0 /ξ0 < 0 in contradiction to the stipulation hξ : H+ → H+ := C \ (H− ∪ iR).

286

S. Feldmann 1.5 "h.dat" "h_xi^eta.dat"

1

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Figure 6. solid line := h(iR), dashed line := hηξ (iR) But, if we replace ω by η, then a positive real approximant is obtained. To get η it is suﬃcient to compute the product Bc (Q)W . The generating function of Bc (Q) yields Bc (Q) = 2ξ1 diag(ξ0 , −1) = diag(0.013, 3.989). In addition, for N = 300 as approximation of 4 +N

1 h(it) 1 dt, W = lim it Q(it) N →∞ 2π −N we get [0.334, 0.997]T such that η := Bc (Q)W becomes [0.005, 3.977]T. To check the positivity of hηξ we represent it as continued fraction: hηξ (z) =

η1 z + η0 = z 2 − ξ1 z − ξ0

1 (az + b) +

. r (η1 z + η0 )

Then the positivity of all coeﬃcients a, b, r, η0 , η1 implies the desired mapping property hηξ : H+ → H+ . From above we know that η0 and η1 are positive. Comparison of the coeﬃcients yields a = 1/η1 ,

b = −(aη0 + ξ1 )/η1 ,

r = −ξ0 − bη0

thus a = 0.251, b = 0.501, and r = 0.001. Figure 6 conﬁrms the positivity of h and hηξ . A mathematical exploration with respect to the preservation of positivity will be provided in a forthcoming paper.

References [1] G.S. Ammar, W.B. Gragg, Numerical experience with a superfast real Toeplitz solver, LAA 34 (1980), 103–116 [2] G.S. Ammar, W.B. Gragg, The generalized Schur algorithm for the superfast solution of Toeplitz systems, in: J. Gilewicz, M. Pidor, W. Siemaszko (Eds.), Rational Approximation and it Applications in Mathematics and Physics, Lecture Notes in Mathematics, vol. 1237, 1987, pp. 315

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[3] G.S. Ammar, W.B. Gragg, Superfast solution of real positive deﬁnite Toeplitz systems, SIAM J. Matrix Anal. Appl.9 (1) (1988), 61–76 [4] A.C. Antoulas, D. Sorenson, S. Gugercin, A survey of model reduction methods for large scale systems, Contemporary Mathematics, AMS Publications (2001), 193–221 [5] M. Van Barel, A. Bultheel, Pad´e techniques for model reduction in linear system theory: a survey, J. Comput. Appl. Math. 14 (1986), 401–438 [6] M. Van Barel, G. Heinig, P. Kravanja, A superfast method for solving Toeplitz linear least squares problems, LAA 366 (2003), 441–457 [7] M. Van Barel, G. Heinig, P. Kravanja, A stabilized superfast solver for nonsymmetric Toeplitz systems, SIAM J. Matrix Anal. Appl.23 (2) (2001), 494–510 [8] O. Brune, Synthesis of a ﬁnite two-terminal network whose driving-point impedance is a prescribed function of frequency, J. of Math. Phys. 10 (1931), 191–236 [9] P.J. Davis, Interpolation & Approximation, Dover, New York, 1975 ¨ [10] L. Fej´er, Uber die Lage der Nullstellen von Polynomen, die aus Minimumforderungen gewisser Art entspringen, Math. Annalen 85 (1922), 41–48 [11] S. Feldmann, Fej´er’s convex hull theorem related to least squares approximation of rational functions, in preparation [12] S. Feldmann, P. Lang, A least squares approach to reduce stable discrete linear systems preserving their stability, LAA 381 (2004), 141–163 [13] S. Feldmann, P. Lang, D. Pr¨atzel-Wolters, A uniﬁed least squares approach to identify and to reduce continuous asymptotically stable systems, LAA 426, Issues 2-3 (2007), 674–689 [14] P.A. Fuhrmann, A Polynomial Approach to Linear Algebra, Springer, New York, Berlin, Heidelberg, 1996 ¨ [15] M. Fujiwara, Uber die algebraischen Gleichungen, deren Wurzeln in einem Kreise oder in einer Halbebene liegen, Math. Z. 24 (1926), 161–169 [16] K.A. Gallivan, S. Thirumalai, P. Van Dooren, V. Vermaut, High performance algorithms for Toeplitz and block Toeplitz matrices, LAA 241-243 (1996), 343–388 [17] I. Gohberg, T. Kailath, V. Olshevsky, Fast Gaussian elimination with partial pivoting for matrices with displacement structure, Math. Comp. 64 (212) (1995), 1557–1576 [18] G.H. Golub, Ch.F. van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, 1991 [19] M. Gu, New fast algorithms for structured least squares problems, SIAM J. Matrix Anal. Appl. 20 (1) (1998), 244–269 [20] M. Gu, Stable and eﬃcient algorithm for structured systems of equations, SIAM J. Matrix Anal. Appl. 19 (2) (1997), 279–306 [21] S. Gugercin, A.C. Antoulas, Model reduction of large-scale systems by least squares, LAA 415, Issue 2-3 (2006), 290–321 [22] G. Heinig, Bezoutiante, Resultante und Spektralverteilungsprobleme f¨ ur Operatorpolynome, Math. Nachr. 91 (1979), 23–43 [23] G. Heinig, U. Jungnickel, Zur L¨ osung von Matrixgleichungen der Form AX − XB = C, Wiss. Zeitschrift der TH Karl-Marx-Stadt 23 (1981), 387–393 [24] G. Heinig, K. Rost, Algebraic Methods for Toeplitz-like Matrices and Operators, Akademie-Verlag, Berlin, 1984

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[25] U. Helmke, P.A. Fuhrmann, Bezoutians, LAA 122-124 (1989), 1039–1097 [26] M.G. Krein, M.A. Naimark, The Method of Symmetric and Hermitian Forms in the Theory of the Separation of the Roots of Algebraic Equations, English translation in Linear and Multilinear Algebra 10 (1981), 265–308 [27] C. Lanczos, Applied Analysis, Prentice-Hall, Inc., Englewood Cliﬀs, New Jersey, 1956 [28] T. Penzl, A cyclic low-rank Smith method for large sparse Lyapunov equations, SIAM J. Sci. Comput., Vol. 21, No. 4 (2000), 1401–1418 [29] W. Rudin, Real and Complex Analysis, Mc Graw-Hill, New York, 1987 [30] J.T. Spanos, M.H. Milman, D.L. Mingori, A new algorithm for L2 -optimal model reduction, Automatica 28 (1992), 897–909 [31] J. Sylvester, On a Theory of the Syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm’s Functions, and that of the greatest Algebraic Common Measure, Philos. Trans. Roy. Soc. London 143 (1853), 407–548 [32] D.A. Wilson, Optimum solution of model reduction problem, Proc. Inst. Elec. Eng. (1970) 1161–1165 [33] H.K. Wimmer, On the history of the Bezoutian and the resultant matrix, LAA 128 (1990) 27–34 [34] W.Y. Yan, J. Lam, An approximative approach to H2 -optimal model reduction, IEEE Trans. Automatic Control, AC-44 (1999), 1341–1358 [35] K. Zhou, J.C. Doyle, K. Glover, Robust and Optimal Control, Prentice Hall, Simon & Schuster, New Jersey, 1995 Sven Feldmann Thiemstrasse 21 D-04299 Leipzig, Germany e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 199, 289–332 c 2010 Birkh¨ auser Verlag Basel/Switzerland

On the Weyl Matrix Balls Corresponding to the Matricial Carath´eodory Problem in Both Nondegenerate and Degenerate Cases Bernd Fritzsche, Bernd Kirstein and Andreas Lasarow Dedicated to the memory of Georg Heinig

Abstract. The main goal of the paper is to determine the Weyl matrix balls associated with an arbitrary matricial Carath´eodory problem. For the special case of a nondegenerate matricial Carath´eodory problem the corresponding Weyl matrix balls were computed by I.V. Kovalishina [Ko] and alternatively by the ﬁrst and the second authors in [FK1, Parts IV and V]. Mathematics Subject Classiﬁcation (2000). Primary: 44A60, 47A57, 30E05 Secondary: 47A56. Keywords. Weyl matrix balls, matricial Carath´eodory problem, matrix polynomials, matricial Carath´eodory functions, matricial Schur functions.

0. Introduction This paper is closely related to the authors’ recent investigations [FKL1] on the matricial Carath´eodory problem in both nondegenerate and degenerate cases. In [FKL1] the authors obtained a parametrization of the solution set of a general (possibly degenerate) matricial Carath´eodory problem in terms of a linear fractional transformation. Using this parametrization we study the set of values of matrices which will be attained by the solutions of the matricial Carath´eodory problem in a prescribed ﬁxed point of the open unit disk. We show that this set of values ﬁlls a (closed) matrix ball and present explicit expressions for the center and the semi-radii of this matrix ball which is also called the Weyl matrix ball associated with the concrete matricial Carath´eodory problem under consideration. Following the classical monograph [Akh] the terminology “Weyl circles” or later The work of the third author of the present paper was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft) on badge LA 1386/2–1.

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“Weyl matrix balls” was consequently used in the Soviet literature (see, e.g., [Or], [KP], [Ko], [Du], and [Mi]). As explained in [Akh, Chapter 1] or in [Ev], the history of the scalar case is intimately related to the classical papers [We], [He], and [Ne]. In the case of a nondegenerate matricial Carath´eodory problem the corresponding Weyl matrix balls were computed by Kovalishina [Ko] and alternatively by the ﬁrst and the second authors in [FK1, Parts IV and V]. In view of the general ˇ theory of matrix balls due to Smuljan [Sm], the semi-radii of a matrix ball are not uniquely determined. In his investigations [Du] on the matricial Schur problem, V.K. Dubovoj has constructed a clever normalization of the left semi-radius. This normalization was adopted to the case of the matricial Carath´eodory problem in [FK1, Parts IV and V]. In the case of the nondegenerate matricial Carath´eodory problem the normalized semi-radii of the Weyl matrix balls can be nicely rewritten with the aid of the Gohberg-Heinig formula (see [GH, Theorem 1.1], for the inverses of block Toeplitz matrices. This will be described at the end of Section 4. In the scalar case the Weyl disks associated with the Carath´eodory problem were already handled by Geronimus (see, e.g., [Ge1], [Ge2], and [Ge3]). A modern view on various aspects of Geronimus’ work was presented in the paper [CG] of Chang and Georgiou who particularly worked out the role of the centers of the Weyl disks in the context of maximum entropy extension. In the paper [FKL2] the authors discussed similar questions and closely related matters for the matrix case of the nondegenerate matricial Carath´eodory problem. The limit behaviour of the normalized semi-radii of the Weyl matrix balls associated with a nondegenerate matrix-valued Carath´eodory function was studied in [FK2]. This paper is organized as follows: In Section 1, after presenting some notations and preliminaries we formulate the main result of this paper (see Theorem 1.1). In Section 2, we recall the parametrization of the solution set of a matricial Carath´eodory problem which was obtained in [FKL1] and we give some additional comments concerning the uniqueness of the functions which appear as parameters in this description. It turns out that much information on the Weyl matrix balls associated with a matricial Carath´eodory problem is contained in a distinguished rational matrix-valued function Θn built from the data. For this reason, the study of this function Θn is one of the central themes of the paper. In Section 3, we introduce the function Θn and show that the restriction of this function onto D is a matrix-valued Schur function. Hereby this Schur function is inner if and only if the considered Carath´eodory problem is nondegenerate. In the particular case that the given sequence (Γj )nj=0 of data satisﬁes det Γ0 = 0 we associate with (Γj )nj=0 a second rational matrix-valued function Θn . The interplay between both functions Θn and Θn is described in Proposition 3.11. In Section 4, we will prove the main result of this paper. More precisely, we will compute the Weyl matrix ball associated with an arbitrary matricial Carath´eodory problem. In Section 5, we consider in detail the case that the given q × q Carath´eodory sequence (Γj )nj=0 satisﬁes det Γ0 = 0. We indicate that the above-mentioned rational matrix-valued function Θn is intimately connected with the so-called reciprocal Carath´eodory sequence (Γj )nj=0 corresponding to (Γj )nj=0 . This enables us to describe the Weyl

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matrix ball associated with (Γj )nj=0 with the aid of Θn (see Theorem 5.2). The main results of Section 6 are Theorem 6.8 and Theorem 6.11 which contain useful recurrence formulas for the functions Θn and Θn , respectively. Finally, in Section 7 we discuss the central q × q Carath´eodory function Ωc,n corresponding to (Γj )nj=0 from the view of Weyl matrix balls.

1. Preliminaries Throughout this paper, let p and q be positive integers. We will use C, N0 , and N to denote the set of all complex numbers, the set of all nonnegative integers, and the set of all positive integers, respectively. If m ∈ N0 and if κ ∈ N0 or κ = ∞, then we will write Nm,κ for the set of all integers k satisfying m ≤ k ≤ κ. The set of all complex p × q matrices will be designated by Cp×q . For each A ∈ Cp×q , let A+ be the Moore-Penrose inverse of A. If A ∈ Cq×q , then det A stands for the determinant of A and tr A denotes the trace of A. Further, for each A ∈ Cq×q , let Re A and Im A be the real part of A and the imaginary part of A, respectively, 1 i.e., Re A := 12 (A + A∗ ) and Im A := 2i (A − A∗ ). The zero matrix which belongs to p×q C will be denoted by 0p×q and the identity matrix which belongs to Cq×q will be designated by Iq . If the size of a zero matrix or a identity matrix is obvious, we will omit the indices. A complex p × q matrix A is said to be contractive if the operator norm of A is not greater than 1. Obviously, a complex p × q matrix A is contractive if and only if the matrix I − A∗ A is nonnegative Hermitian. Let n ∈ N0 . If (Γj )nj=0 is a sequence of complex q × q matrices, then we associate with (Γj )nj=0 the block Toeplitz matrices Sn and Tn given by ⎛ ⎞ Γ0 0 0 ... 0 ⎜ Γ1 Γ0 0 ... 0 ⎟ ⎜ . ⎟ .. ⎜ Γ . .. ⎟ Γ1 Γ0 Sn := ⎜ 2 (1.1) ⎟ ⎜ .. ⎟ .. .. .. ⎝ . . 0 ⎠ . . Γn Γn−1 Γn−2 . . . Γ0 and Tn := Re Sn . of complex q × q matrices is called q × q Carath´eodory sequence A sequence (respectively, nondegenerate q × q Carath´eodory sequence) if the matrix Tn is nonnegative Hermitian (respectively, positive Hermitian). Obviously, if (Γj )nj=0 is a q × q Carath´eodory sequence (respectively, a nondegenerate q × q Carath´eodory sequence) and if m ∈ N0,n , then (Γj )m eodory sequence (rej=0 is also a q × q Carath´ spectively, a nondegenerate q × q Carath´eodory sequence). In addition, a sequence eodory sequence (Γk )∞ k=0 of complex q × q matrices is said to be a q × q Carath´ (respectively, a nondegenerate q × q Carath´eodory sequence) if for every choice of a nonnegative integer m the sequence (Γj )m eodory sequence j=0 is a q × q Carath´ (respectively, a nondegenerate q × q Carath´eodory sequence). (Γj )nj=0

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Let D := {z ∈ C : |z| < 1} and T := {z ∈ C : |z| = 1} be the unit disk and the unit circle of the complex plane, respectively. A complex q × q matrixvalued function Ω : D → Cq×q which is holomorphic in D and for which the real part Re Ω(z) of Ω(z) is nonnegative Hermitian for each z ∈ D is called q × q Carath´eodory function (in D). The set of all q × q Carath´eodory functions (in D) will be denoted by Cq (D). It is a well-known fact that a matrix-valued function Ω : D → Cq×q which is holomorphic in D with Taylor series representation Ω(z) =

∞

Γk z k ,

z ∈ D,

k=0

eodory sequence (see, e.g., belongs to Cq (D) if and only if (Γk )∞ k=0 is a q × q Carath´ [FK1, Section 4 in Part I]). The matricial version of the classical Carath´eodory interpolation problem consists of the following: Let n ∈ N0 and let (Γj )nj=0 be a sequence of complex q × q matrices. Describe the set Cq [D, (Γj )nj=0 ] of all q × q Carath´eodory functions Ω (in D) such that 1 (j) (1.2) Ω (0) = Γj j! holds for each j ∈ N0,n , where Ω(j) (0) is the jth derivative of Ω at the point z = 0. If n ∈ N0 and if (Γj )nj=0 is a sequence of complex q × q matrices, then the set Cq [D, (Γj )nj=0 ] is nonempty if and only if (Γj )nj=0 is a q × q Carath´eodory sequence (see, e.g., [FK1, Section 4 in Part I]). Several approaches to parametrize the solution set Cq [D, (Γj )nj=0 ] of the matricial Carath´eodory problem can be found in the literature (see, e.g., [AK], [Ko], [Dy], [FK1], [BGR], [FF], [Sa], and [FFGK]). An essential common feature is that the discussions are mainly concentrated on the so-called nondegenerate case which is connected with nondegenerate q × q Carath´eodory sequences built from the interpolation data. Nowadays, quite diﬀerent approaches to handle also degenerate cases of matrix interpolation were used (see, e.g., [BH], [BD], [Br], [CH1], [CH2], [DGK2], [Dy, Chapter 7], and [Sa, Chapter 5]). The starting point of the present paper are the descriptions of Cq [D, (Γj )nj=0 ] via linear fractional transformations in the general case, i.e., without additional assumptions, given in [FKL1], where the parameters of the linear fractional transformations are expressed explicitly by the given data of the problem. For the convenience of the reader, we recall the main idea of these parametrizations in the following section (see Theorem 2.1). For this and our forthcoming considerations we need some further notation. If m ∈ N0 , let em,q and εm,q be the matrix polynomials deﬁned by ⎛ m ⎞ z Iq ⎜z m−1 I ⎟ ⎜ .. q ⎟ (1.3) em,q (z) := Iq , zIq , z 2 Iq , . . . , z m Iq and εm,q (z) := ⎜ . ⎟ ⎝ zI ⎠ q Iq

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for all z ∈ C. Let e be a q × q matrix polynomial of degree not greater than m, i.e., there is a complex (m + 1)q × q matrix E such that e(z) = em,q (z)E for each z ∈ C. Then the reciprocal matrix polynomial e˜[m] of e with respect to the unit circle T and the formal degree m is given, for all z ∈ C, by e˜[m] (z) := E ∗ εm,q (z). If n ∈ N0 and if (Γj )nj=0 is a sequence of complex q × q matrices, then let L1 := Re Γ0 ,

R1 := Re Γ0 ,

(1.4)

and moreover (in the case n ≥ 1) we will use for each k ∈ N1,n the notations ⎛ ⎞ Γ1 ⎜ 1 1 ⎜ Γ2 ⎟ ⎟ (1.5) Zk := Γk , Γk−1 , . . . , Γ1 , Yk := ⎜ . ⎟ , 2 2 ⎝ .. ⎠ Γk and

+ + Zk∗ , Rk+1 := Re Γ0 − Yk∗ Tk−1 Yk . (1.6) Lk+1 := Re Γ0 − Zk Tk−1 For each k ∈ N0,n , the matrices Lk+1 and Rk+1 are nonnegative Hermitian if (Γj )nj=0 is a q × q Carath´eodory sequence (see, e.g., [DFK, Lemma 1.1.9]). In the sequel, let n ∈ N0 and let (Γj )nj=0 be a q × q Carath´eodory sequence. In the case n ≥ 1, the sets

Yn := {V ∈ Cnq×q : Tn−1 V = Yn }

and

Zn := {W ∈ Cq×nq : W Tn−1 = Zn }

will be linking elements in the studies below, since the matrix polynomials an , bn , cn , and dn deﬁned by 2 Γ0 if n = 0 an (z) := (1.7) ∗ Γ0 + zen−1,q (z)Sn−1 Vn if n ≥ 1 , 2 Iq if n = 0 (1.8) bn (z) := Iq − zen−1,q (z)Vn if n ≥ 1 , 2 if n = 0 Γ0 cn (z) := (1.9) ∗ Wn Sn−1 zεn−1,q (z) + Γ0 if n ≥ 1 , and 2 Iq if n = 0 (1.10) dn (z) := −Wn zεn−1,q (z) + Iq if n ≥ 1 with some Vn ∈ Yn and Wn ∈ Zn if n ≥ 1 play an essential role in [FKL1]. Note + + that, if n ≥ 1, the matrix Tn−1 Yn belongs to Yn and the matrix Zn Tn−1 belongs to Zn (cf. [FK3, Remark 1.4]). Moreover, in the case n ≥ 1, [FK3, Proposition 2.2] + Yn actually belongs to the set Y n of all Vn ∈ Yn such that det bn implies that Tn−1 + vanishes nowhere in D and from [FK3, Theorem 2.3] one can see that Zn Tn−1 actually belongs to the set Z n of all Wn ∈ Zn such that det dn vanishes nowhere in D, where bn and dn are the matrix polynomials deﬁned by (1.8) and (1.10). A main goal of a parametrization, for every choice of A this paper is to present B w in D, of the set Ω(w) : Ω ∈ Cq [D, (Γj )nj=0 ] in terms of the matrix polynomials an , bn , cn , and dn deﬁned by (1.7), (1.8), (1.9), and (1.10) with some Vn ∈ Y n and

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Wn ∈ Z n if n ≥ 1. We will prove that these sets are matrix balls. Recall that, for each M ∈ Cq×q , each A ∈ Cq×q , and each B ∈ Cq×q , the set K(M ; A, B) of all X ∈ Cq×q which admit a representation X = M + AKB with some contractive complex q × q matrix K is called the matrix ball with center M , left semi-radius A, and right semi-radius B. In order to describe the parameters of the concrete matrix balls in question we introduce the rational matrix-valued function 5 5 ˜[n] Ln+1 + . Θn := Rn+1 b−1 n dn It will turn out that the function Θn is holomorphic in D and that the restriction of Θn onto D belongs to the Schur class Sq×q (D) of all q × q Schur functions (in D). Recall that a matrix-valued function S : D → Cq×q which is holomorphic in D is called q × q Schur function (in D) if at each point z ∈ D the value S(z) of S is a contractive matrix. We will prove that the following theorem holds. Theorem 1.1. Let n ∈ N0 and let (Γj )nj=0 be a q×q Carath´eodory sequence. If n ≥ 1, then let Vn ∈ Y n and let Wn ∈ Z n . Let the matrix polynomials an , bn , cn , and dn be deﬁned by (1.7), (1.8), (1.9), and (1.10). Furthermore, let the complex matrices Ln+1 and Rn+1 be given by (1.4) and (1.6). For each w ∈ D, then by setting 5 5 5 5 −1 [n] + + Θn (w) := Rn+1 bn (w) d˜n (w) Ln+1 , Fn (w) := Ln+1 Θ∗n (w) Rn+1 , 5 5 + and Gn (w) := Ln+1 Θ∗n (w) Rn+1 the identity C D Ω(w) : Ω ∈ Cq [D, (Γj )nj=0 ] = K Mn+1 (w); 2|w|n+1 An+1 (w), Bn+1 (w) is fulﬁlled, where −1 2 [n] (w) (w) + |w| G (w)˜ a (w) , (1.11) c Mn+1 (w) := dn (w) − |w|2 Gn (w)˜b[n] n n n n 7 −1 −1 5 Ln+1 Iq − |w|2 Θ∗n (w)Θn (w) , An+1 (w) := dn (w) and Bn+1(w) :=

7

Iq − |w|2 Θn (w)Θ∗n (w)

−1 5

−1 Rn+1 bn (w)

(1.12)

(1.13)

and where the matrix Mn+1 (w) also admits the representation −1 2 ˜[n] Mn+1 (w) = an (w) + |w|2 c˜[n] (w)F (w) b (w) − |w| (w)F (w) . (1.14) d n n n n n

2. On a parametrization of the solution set Cq [D, (Γj )nj=0 ] In the present section we recall at ﬁrst a description of the set Cq [D, (Γj )nj=0 ] of all solutions of the matricial Carath´eodory problem via linear fractional transformations which is derived in [FKL1]. Moreover, we give some additional comments concerning the uniqueness of the function f ∈ Sq×q (D) appearing as parameter in these linear fractional transformations.

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Theorem 2.1. Let n ∈ N0 and let (Γj )nj=0 be a q × q Carath´eodory sequence. If n ≥ 1, then let Vn ∈ Y n and Wn ∈ Z n . Let the matrix polynomials an , bn , cn , and dn be given by (1.7), (1.8), (1.9), and (1.10). Furthermore, let the matrices Ln+1 and Rn+1 be deﬁned by (1.4) and (1.6). 5 5 + [n] (a) If f ∈ Sq×q (D), then the matrices z d˜n (z) Ln+1 f (z) Rn+1 + bn (z) and 5 5 + [n] z Ln+1 f (z) Rn+1 ˜bn (z) + dn (z) are nonsingular for each z ∈ D and the function Ω : D → Cq×q given by −1 Ω(z) := −z˜ c[n] z d˜n[n] (z)F (z) + bn (z) (2.1) n (z)F (z) + an (z) belongs to Cq [D, (Γj )nj=0 ] and satisﬁes, for each z ∈ D, the representation −1 [n] −zG(z)˜ a Ω(z) = zG(z)˜b[n] (z) + d (z) (z) + c (z) , (2.2) n n n n 5 5 5 + 5 + where F := Ln+1 f Rn+1 and G := Ln+1 f Rn+1 . (b) If Ω ∈ Cq [D, (Γj )nj=0 ], then there is an f ∈ Sq×q (D) so that Ω admits, for each z ∈ D, the representations −1 [n] ˜ Ω(z) = −z˜ c[n] (z)F (z) + a (z) z d (z)F (z) + b (z) n n n n 5 5 5 +5 + and (2.2), where F := Ln+1 f Rn+1 and G := Ln+1 f Rn+1 . A proof of Theorem 2.1 is given by [FKL1, Theorems 3.2 and 3.7]. Contrary to the well-known circumstance in the nondegenerate case (see, e.g., [AK], [Ko], [Dy], [BGR], [FF], and [FK1]), the underlying function f ∈ Sq×q (D) by the linear fractional transformations stated in (2.1) and (2.2), respectively, is not uniquely determined via f in general. The following results clarify this fact. Proposition 2.2. Let n ∈ N0 and let (Γj )nj=0 be a q × q Carath´eodory sequence. If n ≥ 1, then let Vn ∈ Y n and let Wn ∈ Z n . Let the matrix polynomials an , bn , cn , and dn be deﬁned by (1.7), (1.8), (1.9), and (1.10). Let the matrices Ln+1 and Rn+1 be given by (1.4) and (1.6). Further, for j ∈ {1, 2}, let fj be a q × q Schur 5 + 5 function (in D), let Fj := Ln+1 fj Rn+1 , and let Ωj : D → Cq×q be deﬁned by −1 Ωj (z) := −z˜ c[n] z d˜n[n] (z)Fj (z) + bn (z) . n (z)Fj (z) + an (z) (a) Let w ∈ D \ {0}. Then the following statements are equivalent: (i) Ω1 (w) = Ω2 (w). + + + (ii) Ln+1 L+ n+1 f1 (w)Rn+1 Rn+1 = Ln+1 Ln+1 f2 (w)Rn+1 Rn+1 . (b) The following statements are equivalent: (iii) Ω1 = Ω2 . + + + (iv) Ln+1 L+ n+1 f1 Rn+1 Rn+1 = Ln+1 Ln+1 f2 Rn+1 Rn+1 . Proof. For j ∈ {1, 2}, we also use the setting 5 5 + Gj := Ln+1 fj Rn+1 .

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First we prove that (a) holds. Let w ∈ D \ {0}. (i) ⇒ (ii): Suppose that (i) holds. Thus, part (a) of Theorem 2.1 yields −1 [n] −wG wG1 (w)˜b[n] (w) + d (w) (w)˜ a (w) + c (w) n 1 n n n −1 [n] ˜ (w)F (w) + a (w) w d (w)F (w) + b (w) = −w˜ c[n] 1 n 1 n n n −1 [n] [n] = −w˜ cn (w)F2 (w) + an (w) wd˜n (w)F2 (w) + bn (w) and consequently ˜[n] a[n] a[n] − w2 G1 (w)˜ n (w)dn (w)F2 (w) + cn (w)bn (w) − wG1 (w)˜ n (w)bn (w) [n] ˜ + wc (w) d (w)F (w) n 2 n [n] ˜ a[n] (w) + c (w) w d (w)F (w) + b (w) = −wG1 (w)˜ n 2 n n n [n] [n] ˜ = wG1 (w)bn (w) + dn (w) −w˜ cn (w)F2 (w) + an (w) = − w2 G1 (w)˜b[n] (w)˜ c[n] (w)F2 (w) + dn (w)an (w) + wG1 (w)˜b[n] (w)an (w) n

n

n

c[n] − wdn (w)˜ n (w)F2 (w).

(2.3)

Moreover, from [FK3, Theorems 1.7 and 2.3] and [DFK, Lemma 1.2.2] we get ˜[n] ˜[n] dn (w)an (w) = cn (w)bn (w) and a ˜[n] c[n] (2.4) n (w)dn (w) = bn (w)˜ n (w). By virtue of [FKL1, part (b) of Proposition 2.4] we have a ˜[n] (w)bn (w) + ˜b[n] (w)an (w) = 2wn Rn+1 , n

cn (w)d˜n[n] (w)

n

dn (w)˜ c[n] n (w)

n

+ = 2w Ln+1 . Using (2.4), (2.5), and (2.6) from (2.3) we can conclude

(2.5) (2.6)

2wn+1 G1 (w)Rn+1 = 2wn+1 Ln+1 F2 (w). 5 + Multiplying the last equation from the left by 2w1n+1 Ln+1 and from the right 5 + by Rn+1 we get the identity stated in (ii). (ii) ⇒ (i): From (ii) it follows F1 (w) = F2 (w) and hence (i). Therefore, part (a) is veriﬁed. Part (b) is then an easy consequence of (a) and a continuity argument (note part (a) of Theorem 2.1). Remark 2.3. Let the assumptions of Proposition 2.2 be fulﬁlled. Then a combination of Theorem 2.1 and part (b) of Proposition 2.2 shows that there is a bijective correspondence between the solution set Cq [D, (Γj )nj=0 ] and the set of all + matrix-valued functions f ∈ Sq×q (D) satisfying Ln+1 L+ n+1 f Rn+1 Rn+1 = f . Remark 2.4. Let the assumptions of Proposition 2.2 be fulﬁlled. Furthermore, for 5 5 + j ∈ {1, 2}, let Gj := Ln+1 fj Rn+1 . In view of Theorem 2.1 we mention that from Proposition 2.2 one can see that the following statements are equivalent: (i) Ω1 = Ω2 . (ii) F1 = F2 . (iii) G1 = G2 . (iv) Ln+1 f1 Rn+1 = Ln+1f2 Rn+1 .

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3. Particular matrix-valued Schur functions associated with given q × q Carath´eodory sequences In this section, we will discuss particular rational matrix-valued functions the restriction of which onto D are q×q Schur functions (in D) and which are constructed from given ﬁnite q × q Carath´eodory sequences based on the complex q × q matrix polynomials an , bn , cn , and dn deﬁned by (1.7), (1.8), (1.9), and (1.10). For this reason, we ﬁrstly prove certain identities for these matrix polynomials. If G is a nonempty subset of C and if e : G → Cq×q is a matrix-valued function, then let Ne := {w ∈ G : det e(w) = 0}. Remark 3.1. Let n ∈ N0 and let (Γj )nj=0 be a q×q Carath´eodory sequence. If n ≥ 1, then let Vn ∈ Yn and let Wn ∈ Zn . Furthermore, let the matrix polynomials an , bn , cn , and dn be deﬁned by (1.7), (1.8), (1.9), and (1.10). In view of bn (0) = I, dn (0) = I, and [DFK, Lemma 1.2.2] one can immediately conclude that the set N := Nbn ∪N˜b[n] ∪Ndn ∪Nd˜[n] consists of at most 4nq complex numbers. Similarly, n n if the matrix Γ0 is nonsingular, then from an (0) = Γ0 , cn (0) = Γ0 , and [DFK, Lemma 1.2.2] one can see that the set M := Nan ∪ Na˜[n] ∪ Ncn ∪ Nc˜[n] consists of n n at most 4nq complex numbers. Lemma 3.2. Let n ∈ N0 and let (Γj )nj=0 be a q × q Carath´eodory sequence. Let the matrices Ln+1 and Rn+1 be given by (1.4) and (1.6). Furthermore, if n ≥ 1, then let Vn ∈ Yn and let Wn ∈ Zn . (a) The matrix polynomials bn and dn deﬁned by (1.8) and (1.10) satisfy for each w ∈ C \ N the identity [n] ˜b (w) −1 Rn+1 bn (w) −1 = dn (w) −1 Ln+1 d˜[n] (w) −1 , (3.1) n n where N := Nbn ∪ N˜b[n] ∪ Ndn ∪ Nd˜[n] . n n (b) Let the matrix polynomials an and cn be deﬁned by (1.7) and (1.9). Suppose that the set M := Nan ∪ Na˜[n] ∪ Ncn ∪ Nc˜[n] does not coincide with C. Then n n [n] −1 −1 −1 −1 a ˜n (w) Rn+1 an (w) = cn (w) Ln+1 c˜[n] (3.2) n (w) is satisﬁed for each C \ M. Proof. We know from Remark 3.1 that N consists of at most 4nq complex numbers. Applying [FK3, Theorems 1.7 and 2.3] and [DFK, Lemma 1.2.2] we get (2.4) for each w ∈ C. From [FKL1, part (b) of Proposition 2.4] we see that the identities (2.5) and (2.6) are satisﬁed for each w ∈ C. Since (2.4), (2.5), and (2.6) imply ˜b[n] (w)an (w) ˜[n] 2wn Rn+1 = a n (w)bn (w) + n −1 −1 ˜[n] c[n] bn (w) + ˜b[n] = ˜b[n] n (w)˜ n (w) dn (w) n (w) dn (w) cn (w)bn (w) −1 [n] −1 ˜[n] dn (w)˜ = ˜b[n] c[n] d˜n (w) bn (w) n (w) dn (w) n (w) + cn (w)dn (w) [n] −1 −1 n˜[n] Ln+1 d˜n (w) bn (w) (3.3) = 2w bn (w) dn (w)

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for each w ∈ C \ (Ndn ∪ Nd˜[n] ), we can infer that the equality (3.1) is satisﬁed n for each w ∈ C \ (N ∪ {0}). Finally, a continuity argument yields that (3.1) holds actually for every choice of w in C \ N. Now suppose that M = C. From the ﬁrst equation in (3.3), (2.4), and (2.6), we obtain then −1 [n] −1 ˜[n] 2wn Rn+1 = a ˜[n] dn (w)an (w) + a ˜[n] ˜n (w) an (w) n (w) cn (w) n (w)dn (w) c −1 [n] −1 [n] [n] [n] ˜ dn (w)˜ = a ˜n (w) cn (w) cn (w) + cn (w)dn (w) c˜n (w) an (w) [n] −1 −1 n [n] ˜n (w) cn (w) Ln+1 c˜n (w) an (w) = 2w a for each w ∈ C\ M. By continuity, (3.2) follows for every choice of w in C\ M.

Remark 3.3. Let n ∈ N0 and let (Γj )nj=0 be a q×q Carath´eodory sequence. If n ≥ 1, then let Vn , Vn ∈ Yn and let Wn , Wn ∈ Zn . Let an , bn , cn and dn be the matrix polynomials which are deﬁned by (1.7), (1.8), (1.9), and (1.10). Furthermore, let an , bn , cn , and dn be the matrix polynomials which are deﬁned analogously as an , bn , cn , and dn using the matrices Vn and Wn instead of Vn and Wn , respectively. From the deﬁnition of these matrix polynomials and Lemma 3.2 one can see that [n] [n] ˜b (w) −1 Rn+1 bn (w) −1 = b ˜ (w) −1 Rn+1 bn (w) −1 n n for all w ∈ C \ (Nbn ∪ N˜b[n] ∪ Nbn ∪ Nb˜ [n] ) and that n n [n] [n] −1 −1 −1 ˜ (w) −1 dn (w) Ln+1 d˜n (w) = dn (w) Ln+1 d n for all w ∈ C \ (Ndn ∪ Nd˜[n] ∪ Ndn ∪ Nd˜ [n] ) hold, where the matrices Ln+1 and n n Rn+1 are given by (1.4) and (1.6). Similarly, if N1 := Nan ∪ Na˜[n] ∪ Nan ∪ Na˜[n] n n does not coincide with C and if w ∈ C \ N1 , then [n] −1 −1 [n] −1 −1 ˜n (w) a ˜n (w) Rn+1 an (w) = a Rn+1 an (w) and if N2 := Ncn∪Nc˜[n]∪Ncn∪N˜c[n] does not coincide with C and if w ∈ C\N2 , then n n [n] −1 −1 −1 −1 cn (w) Ln+1 c˜n (w) = cn (w) Ln+1 ˜c[n] . n (w) For every positive real number ρ let K(0; ρ) := {w ∈ C : |w| < ρ}. Proposition 3.4. Let n ∈ N0 and let (Γj )nj=0 be a q × q Carath´eodory sequence. If n ≥ 1, then let Vn ∈ Y n and let Wn ∈ Z n . Let the q × q matrix polynomials bn and dn be given by (1.8) and (1.10). Furthermore, let the matrices Ln+1 and Rn+1 be deﬁned by (1.4) and (1.6). Then there is a real number ρ > 1 such that the rational matrix-valued function 5 5 ˜[n] Ln+1 + Θn := Rn+1 b−1 (3.4) n dn is holomorphic in K(0; ρ). Moreover, Θn admits the representation 5 5 + −1 Ln+1 , Θn = Rn+1 ˜b[n] n dn

(3.5)

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for each w ∈ D the inequalities + Θn (w)Θ∗n (w) ≤ Rn+1 Rn+1

and

Θ∗n (w)Θn (w) ≤ Ln+1 L+ n+1

(3.6)

are fulﬁlled, and for each z ∈ T the equations + Θn (z)Θ∗n (z) = Rn+1 Rn+1

and

Θ∗n (z)Θn (z) = Ln+1 L+ n+1

(3.7)

hold. In particular, the restriction of Θn onto D is a q × q Schur function (in D). Proof. Because of the choice of Vn and Wn (in the case n ≥ 1), for each w ∈ D the matrices bn (w) and dn (w) are both nonsingular and 5 5 −1 [n] + Θn (w) = Rn+1 bn (w) d˜n (w) Ln+1 holds. Using Lemma 3.2 we get then 5 5 [n] −1 −1 [n] + + ˜ Θn (w) = d˜n (w) Ln+1 Rn+1 ˜b[n] Rn+1 bn (w) n (w) bn (w) 5 5 −1 −1 [n] + + = Rn+1 ˜b[n] Ln+1 d˜n[n] (w) d˜n (w) Ln+1 n (w) dn (w) 5 −1 5 + = Rn+1 ˜b[n] (w) dn (w) Ln+1 n

for every choice of w in D. Hence (3.5) follows by a continuity argument. According to Remark 3.1 the set N := Nbn ∪N˜b[n] ∪Ndn ∪Nd˜[n] consists of at most 4nq complex n n numbers. Because of (3.5), [DFK, Lemma 1.2.2], and Lemma 3.2 we obtain 5 −1 −∗ [n] ∗ 5 + + ˜b (z) Rn+1 ˜b[n] Ln+1 dn (z) Rn+1 Θn (z)Θ∗n (z) = n (z) dn (z) n 5 5 −1 −1 + + = Rn+1 ˜b[n] Ln+1 d˜n[n] (z) bn (z) Rn+1 n (z) dn (z) 5 5 + + + = Rn+1 Rn+1 Rn+1 = Rn+1 Rn+1 for each z ∈ T \ N. Based on (3.4) one gets analogously Θ∗n (z)Θn (z) = Ln+1 L+ n+1 for every choice of z in T \ N. Hence the rational matrix-valued function Θn fulﬁlls the identities in (3.7) actually for each z ∈ T and there is a real number ρ > 1 such that Θn is holomorphic in K(0; ρ). In particular, from Ln+1 L+ n+1 ≤ I we see that for each z ∈ T the matrix Θn (z) is contractive. Thus the maximum modulus principle for holomorphic functions yields that the restriction of Θn onto D belongs to Sq×q (D). Let w ∈ D. Multiplying the inequality Θ∗n (w)Θn (w) ≤ I from the left and from the right by Ln+1 L+ n+1 and using (3.5) we get the second inequality in (3.6). The ﬁrst one follows analogously. Corollary 3.5. Let the assumptions of Proposition 3.4 be fulﬁlled. Then the following statements are equivalent: (i) There is a unique Ω ∈ Cq (D) such that (1.2) holds for each j ∈ N0,n . (ii) Θn (w) = 0 for each w ∈ C. (iii) There is some z0 ∈ T such that Θn (z0 ) is strictly contractive. Proof. The matrix Ln+1 L+ n+1 is strictly contractive if and only if Ln+1 = 0. Thus Proposition 3.4 and [FKL1, Lemma 6.1] yield the asserted equivalences.

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In the next result, we will analyze the construction of a q × q Schur function given by Proposition 3.4 for the case that the underlying q × q Carath´eodory sequence (Γj )nj=0 is nondegenerate. Recall ﬁrstly that, by a q × q Blaschke-Potapov elementary factor (with respect to the signature matrix Iq ) with zero at w0 ∈ D we mean a rational complex q × q matrix-valued function Bw0 which is given via (3.8) Bw0 = V1 Iq + (bw0 − 1)P V2 , where V1 and V2 are some unitary q × q matrices, where P is some non-zero idempotent and Hermitian q × q matrix, and where bw0 is the elementary Blaschke factor corresponding to w0 , i.e., bw0 is the rational function deﬁned by ⎧ w for each w ∈ C if w0 = 0 ⎨ C D bw0 (w) := w0 w0 − w ⎩ for each w ∈ C \ w10 if w0 = 0 . |w0 | 1 − w0 w The notion ﬁnite q × q Blaschke-Potapov product (with respect to Iq ) will be used to denote a constant unitary q×q matrix-valued function or a ﬁnite product of q×q Blaschke-Potapov elementary factors deﬁned as in (3.8). Moreover, we note that each q × q Schur function f (in D) has radial boundary values λ-almost everywhere on T, where λ stands for the linear Lebesgue-Borel measure on T. If f is a q × q Schur function in D such that its radial boundary values are unitary λ-almost everywhere on T, then f is called an inner q × q Schur function (in D). Corollary 3.6. Let the assumptions of Proposition 3.4 be fulﬁlled. Then the following statements are equivalent: (i) (ii) (iii) (iv) (v)

The q × q Carath´eodory sequence (Γj )nj=0 is nondegenerate. For each z ∈ T the matrix Θn (z) is unitary. There is some w ∈ D ∪ T such that the matrix Θn (w) is nonsingular. The restriction of Θn onto D is an inner q × q Schur function. Θn is a ﬁnite Blaschke-Potapov product (with respect to Iq ).

Proof. Proposition 3.4 yields that the identities in (3.7) hold for each z ∈ T, that the inequalities in (3.6) are satisﬁed for each w ∈ D, and that the restriction of Θn onto D belongs to Sq×q (D). Thus the equivalence of (i), (ii), (iii), and (iv) follows from [FKL1, Lemma 5.1], whereas the equivalence of (iv) and (v) is a consequence of [FFK, Corollary 13 and Proposition 31]. In the second part of this section we present now a similar construction of a q × q Schur function as in Proposition 3.4, where the matrix polynomials an and cn deﬁned by (1.7) and (1.9) are involved instead of the matrix polynomials bn and dn deﬁned by (1.8) and (1.10). Note that, in view of (1.7) and (1.9), it is not hard to accept that one has to take for granted then a given q × q Carath´eodory sequence (Γj )nj=0 with the additional condition that the matrix Γ0 is nonsingular. Remark 3.7. Let n ∈ N0 and let (Γj )nj=0 be a q × q Carath´eodory sequence with nonsingular matrix Γ0 . If n ≥ 1, then let Vn ∈ Y n and let Wn ∈ Z n . From [FK1,

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Remark 30 in Part V] and [FK3, Remark 1.1, Theorem 1.7, and Theorem 2.3] one can see that the functions det an and det cn vanish nowhere in D. Proposition 3.8. Let n ∈ N0 and let (Γj )nj=0 be a q × q Carath´eodory sequence with nonsingular matrix Γ0 . If n ≥ 1, then let Vn ∈ Y n and let Wn ∈ Z n . Let the matrix polynomials an and cn be deﬁned by (1.7) and (1.9). Furthermore, let the matrices Ln+1 and Rn+1 be given by (1.4) and (1.6). Then there is a real number ρ > 1 such that the matrix-valued function 5 5 + ˜[n] Ln+1 (3.9) Θn := Rn+1 a−1 n c n is holomorphic in K(0; ρ). Moreover, Θn admits the representation 5 + [n] −1 5 Θn = Rn+1 a ˜n cn Ln+1 , for each w ∈ D the inequalities ∗ + Θn (w) Θn (w) ≤ Rn+1 Rn+1

and

(3.10)

∗ Θn(w) Θn (w) ≤ Ln+1L+ n+1

are fulﬁlled, and for each z ∈ T the equations ∗ ∗ + Θn (z) Θn (z) = Rn+1 Rn+1 and Θn(z) Θn (z) = Ln+1 L+ n+1 hold. In particular, the restriction of Θn onto D is a q × q Schur function (in D). Proof. Because of Remark 3.7 the functions det an and det cn vanish nowhere in D. Consequently, the relation 5 5 −1 [n] + c˜n (w) Ln+1 Θn (w) = Rn+1 an (w) is satisﬁed for each w ∈ D. Using Remark 3.1 and part (b) of Lemma 3.2 we get then (3.10) and that the set M := Nan ∪ Na˜[n] ∪ Ncn ∪ Nc˜[n] consists of at most n n 4nq complex numbers. The rest of the assertion can be veriﬁed analogously to the given proof of Proposition 3.4. Corollary 3.9. Let the assumptions of Proposition 3.8 be fulﬁlled. Then the following statements are equivalent: (i) There is a unique Ω ∈ Cq (D) such that (1.2) holds for each j ∈ N0,n . (ii) Θn (w) = 0 for each w ∈ C. (iii) There is some z0 ∈ T such that the matrix Θn (z0 ) is strictly contractive. Proof. Use Proposition 3.8, [FKL1, Lemma 6.1], and the fact that the matrix Ln+1 L+ n+1 is strictly contractive if and only if Ln+1 = 0. Corollary 3.10. Let the assumptions of Proposition 3.8 be fulﬁlled. Then the following statements are equivalent: (i) The q × q Carath´eodory sequence (Γj )nj=0 is nondegenerate. (ii) For each z ∈ T the matrix Θn (z) is unitary. (iii) There is some w ∈ D ∪ T such that the matrix Θn (w) is nonsingular. (iv) The restriction of Θn onto D is an inner q × q Schur function. (v) Θn is a ﬁnite Blaschke-Potapov product (with respect to Iq ).

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Proof. Using Proposition 3.8, [FKL1, Lemma 5.1], and [FFK, Corollary 13 and Proposition 31] the proof is analogous to the proof of Corollary 3.6. Note that from Corollary 3.5 and Corollary 3.9 we see particularly that, under the assumptions of Proposition 3.8, if there is a unique Ω ∈ Cq (D) such that (1.2) is satisﬁed for each j ∈ N0,n , then the rational matrix-valued functions Θn and Θn deﬁned by (3.4) and (3.9) coincide. Generally, the following connection between both functions is available. Proposition 3.11. Let n ∈ N0 and let (Γj )nj=0 be a q × q Carath´eodory sequence with nonsingular matrix Γ0 . If n ≥ 1, then let Vn ∈ Y n and let Wn ∈ Z n . Let the matrix polynomials an , bn , cn , and dn be deﬁned by (1.7), (1.8), (1.9), and (1.10). Furthermore, let the matrices Ln+1 and Rn+1 be given by (1.4) and (1.6) and let the matrix-valued functions Θn and Θn be given by (3.4) and (3.9). For every choice of w in D, then the identities 5 −1 −1 5 Ln+1 (3.11) Θn (w) + Θn (w) = 2wn Rn+1 an (w) dn (w) and

Θn (w) + Θn (w) = 2wn are fulﬁlled.

5 −1 −1 5 Rn+1 bn (w) Ln+1 cn (w)

(3.12)

Proof. Obviously, the complex-valued functions det bn and det dn vanish nowhere in D. Remark 3.7 shows that det an and det cn vanish nowhere in D as well. Let w ∈ D. Using [FK3, Remark 1.1, Theorem 1.7, and Theorem 2.3] we get 5 −1 −1 5 + [n] Θn (w) = Rn+1 a ˜n (w) cn (w) dn (w) dn (w) Ln+1 5 −1 −1 5 + [n] dn (w) = Rn+1 a ˜n (w)bn (w) an (w) Ln+1 (3.13) and, in view of (3.9), similarly 5 5 −1 −1 + dn (w)˜ c[n] cn (w) Θn (w) = Rn+1 bn (w) n (w) Ln+1 .

(3.14)

Moreover, from [FKL1, part (b) of Proposition 2.4] we obtain −1 −1 = 2wn Rn+1 an (w) − ˜b[n] a ˜[n] n (w)bn (w) an (w) n (w)

(3.15)

and

−1 −1 n cn (w) dn (w)˜ c[n] Ln+1 − d˜n[n] (w). (3.16) n (w) = 2w cn (w) Because of (3.4), (3.13), and (3.15) it follows (3.11). Analogously, (3.12) is a consequence of (3.5), (3.14), and (3.16). Corollary 3.12. Let the assumptions of Proposition 3.11 be fulﬁlled. Then there is a real number ρ > 1 such that the matrix-valued function 5 5 −1 Θn := Rn+1 a−1 Ln+1 (3.17) n dn is holomorphic in K(0; ρ). Moreover, Θn admits the representation 5 5 −1 Θn = Rn+1 b−1 Ln+1 n cn and the restriction of Θn onto D is a q × q Schur function (in D).

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Proof. Use Proposition 3.11 in combination with Proposition 3.4, Proposition 3.8, and a matricial version of Schwarz lemma (see, e.g., [DFK, Lemma 2.3.1]).

4. On the Weyl matrix balls associated with matricial Carath´eodory sequences In this section, we will prove Theorem 1.1 which describes the Weyl matrix balls of the solutions of the matricial Carath´eodory problem in both nondegenerate and degenerate cases. Moreover, we discuss the particular choice of the semi-radii of the matrix balls in question and we compare Theorem 1.1 with the corresponding result for the nondegenerate case. First of all, we remark the following. Remark 4.1. Let n ∈ N0 and let (Γj )nj=0 be a q × q Carath´eodory sequence. If n ≥ 1, then let Vn ∈ Y n and let Wn ∈ Z n . Let the matrix polynomials an , bn , cn , and dn be deﬁned by (1.7), (1.8), (1.9), and (1.10). Furthermore, let the matrixvalued function Θn be deﬁned by (3.4). For each w ∈ D, from Proposition 3.4 and [FKL1, Lemma 3.1] one can see that the matrices Iq − |w|2 Θ∗n (w)Θn (w) and Iq − |w|2 Θn (w)Θ∗n (w) are both positive Hermitian and that the matrices 5 5 + dn (w) − |w|2 Ln+1 Θ∗n (w) Rn+1 ˜b[n] n (w) and

5 5 + bn (w) − |w|2 d˜n[n] (w) Ln+1 Θ∗n (w) Rn+1

are both nonsingular. We start now with our proof of Theorem 1.1. Proof of Theorem 1.1. Let w ∈ D. Subject to Remark 4.1 the matrices Mn+1 (w), An+1 (w), and Bn+1 (w) are well deﬁned and representation (1.14) of Mn+1 (w) is an immediate consequence of [FKL1, Lemma 3.1]. In view of Theorem 2.1, if f ∈ Sq×q (D), then we use in the following the setting Ωf : D → Cq×q deﬁned by −1 ˜[n] (z)F (z) + bn (z) c[n] (z)F (z) + a (z) z d , (4.1) Ωf (z) := −z˜ n n n where F :=

+ 5 Ln+1 f Rn+1 . Therefore, Theorem 2.1 yields C D C D Ω(w) : Ω ∈ Cq [D, (Γj )nj=0 ] = Ωf (w) : f ∈ Sq×q (D) .

5

(4.2)

Let u ∈ T be such that the relation w = |w|u is satisﬁed, let E := wΘ∗n (w), and let 5 5 + [n] Φ := dn (w) − |w|2 Ln+1 Θ∗n (w) Rn+1 ˜bn (w), where Θn is the rational matrixvalued function deﬁned by (3.4). Remark 4.1 shows that the matrix E is strictly contractive and that the matrix Φ is nonsingular. Moreover, Proposition 3.4 and the choice of Vn and Wn (in the case n ≥ 1) provide us that 5 5 5 −1 [n] −15 + + Ln+1 . d˜n (w) Ln+1 = Rn+1 ˜b[n] Θn (w) = Rn+1 bn (w) n (w) dn (w)

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Therefore, a straightforward calculation yields 5 −1 5 Ln+1 (I − EE ∗ ) = Φ dn (w) Ln+1 and hence

5 −1 5 Ln+1 (I − EE ∗ )−1 . (4.3) Φ−1 Ln+1 = dn (w) Furthermore, from [FKL1, Proposition 2.4] we get (2.5) and (2.6). Because of [FK3, Theorems 1.7 and 2.3] and [DFK, Lemma 1.2.2] we have (2.4). The following considerations of the proof are divided into three steps. Step A. In this ﬁrst step we consider an arbitrary f ∈ Sq×q (D). Using (4.1) with 5 + 5 F := Ln+1 f Rn+1 , we are going to prove that Ωf (w) − Mn+1 (w)

√ −1√ −1 = −2wn+1An+1(w) I −EE ∗ f (w) + E I + E ∗f (w) I −E ∗E Bn+1(w) (4.4)

holds. For this reason, ﬁrst we note that Theorem 2.1 yields that the matrix 5 5 + Ψf := wd˜n[n] (w) Ln+1 f (w) Rn+1 + bn (w) (4.5) is nonsingular. Because of (1.11), (4.1), (2.4), (2.5), and (2.6) we obtain then Φ Ωf(w) − Mn+1 (w) Ψf 5 5 + = dn (w) − |w|2 Ln+1 Θ∗n (w) Rn+1 ˜b[n] −w˜ c[n] n (w) n (w)F (w) + an (w) 5 5 + [n] − cn (w) + |w|2 Ln+1 Θ∗n (w) Rn+1 a ˜n (w) wd˜n[n] (w)F (w) + bn (w) 5 5 + 2 c[n] Ln+1 Θ∗n (w) Rn+1 ˜b[n] = − wdn (w)˜ n (w)F (w) − |w| n (w)an (w) 5 5 + [n] 2 ∗ ˜ + dn (w)an (w) + w|w| Ln+1 Θn (w) Rn+1 bn (w)˜ c[n] n (w)F (w) 5 5 + [n] 2 ∗ [n] ˜ ˜n (w)bn (w) − wcn (w)dn (w)F (w) − |w| Ln+1Θn (w) Rn+1 a 5 5 + [n] 2 ∗ ˜n (w)d˜n[n] (w)F (w) − cn (w)bn (w) − w|w| Ln+1 Θn (w) Rn+1 a 5 5 5 5 + + n+1 n 2 ∗ = − 2w L L f (w) R − 2w |w| L Θ (w) Rn+1 Rn+1 n+1 n+1 n+1 n+1 n 5 5 = − 2wn+1 Ln+1 f (w) + E Rn+1 and consequently

5 5 Rn+1 Ψ−1 Ωf (w) − Mn+1 (w) = −2wn+1 Φ−1 Ln+1 f (w) + E f .

(4.6)

Moreover, from (3.4), Vn ∈ Y n , Wn ∈ Z n , and (4.5) we can conclude that 5 I + E ∗ f (w) Rn+1 5 5 = Rn+1 + wΘn (w)f (w) Rn+1 5 5 5 5 −1 −1 [n] + d˜n (w) Ln+1 f (w) Rn+1 = Rn+1 bn (w) bn (w) + w Rn+1 bn (w) 5 −1 = Rn+1 bn (w) Ψf . (4.7) Since the matrix E is strictly contractive, the matrix I + E ∗ f (w) is nonsingular (see, e.g., [DFK, Lemma 1.1.12, Lemma 1.1.13, and Remark 1.1.2]). Therefore, from (4.7) we get the identity 5 −1 5 −1 ∗ Rn+1 Ψ−1 Rn+1 bn (w) . (4.8) f = I + E f (w)

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Combining (4.6), (4.3), (4.8), (1.12), and (1.13) we obtain Ωf (w) − Mn+1 (w) −1 5 −1 = − 2wn+1 dn (w) Ln+1 I − EE ∗ f (w) + E −1 5 −1 Rn+1 bn (w) · I + E ∗ f (w) √ −1 5 −1 √ −1 = − 2wn+1 dn (w) Ln+1 I − EE ∗ I − EE ∗ f (w) + E √ −1 √ −1 −1 5 · I + E ∗ f (w) I − E∗E I − E∗E Rn+1 bn (w) √ √ −1 −1 = − 2wn+1An+1 (w) I −EE ∗ I −E ∗ E Bn+1 (w). f (w) + E I + E ∗ f (w) Thus (4.4) is proved. Step B. We are going to check that C D Ω(w) : Ω ∈ Cq [D, (Γj )nj=0 ] ⊆ K Mn+1 (w); 2|w|n+1 An+1 (w), Bn+1 (w) (4.9) A B is satisﬁed. Let X ∈ Ω(w) : Ω ∈ Cq [D, (Γj )nj=0 ] . According to (4.2) there is an f ∈ Sq×q (D) such that X = Ωf (w), where Ωf : D → Cq×q is given by (4.1) with 5 + 5 F := Ln+1 f Rn+1 . By virtue of Step A we have then (4.4). Since the matrix √ √ −1 −1 I − EE ∗ I − EE ∗ E H(E) := √ √ −1 −1 I − E∗E E∗ I − E∗E fulﬁlls the identity

∗ Iq H(E) 0

I 0 H(E) = q −Iq 0

0 −Iq

(4.10)

(see, e.g., [DFK, Lemma 3.6.32 and Lemma 1.1.12]), an application of [DFK, part (a) of Theorem 1.6.1] provides us that the complex q × q matrix √ −1 √ −1 I − EE ∗ I − E∗E f (w) + E I + E ∗ f (w) is contractive. Since u belongs to T, the matrix √ −1 √ −1 K := −un+1 I − EE ∗ I − E ∗E f (w) + E I + E ∗ f (w) is also contractive and because of (4.4) we have ﬁnally X − Mn+1 (w) = Ωf (w) − Mn+1 (w) = 2|w|n+1 An+1 (w)KBn+1 (w). Consequently, (4.9) is proved. Step C. We are going to check that C D K Mn+1 (w); 2|w|n+1 An+1 (w), Bn+1 (w) ⊆ Ω(w) : Ω ∈ Cq [D, (Γj )nj=0 ] (4.11) is satisﬁed as well. For this reason, we consider an arbitrary element X of the matrix ball K Mn+1 (w); 2|w|n+1 An+1 (w), Bn+1 (w) . Then there is a contractive q × q matrix C such that X − Mn+1 (w) = −2wn+1 An+1 (w) −un+1 C Bn+1 (w)

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holds. According to (4.10) and [DFK, part (a) of Theorem 1.6.2] there is a contractive q × q matrix D such that √ √ −1 −un+1 C = I − EE ∗ (D + E)(I + E ∗ D)−1 I − E ∗ E. Let f be the constant matrix-valued function deﬁned on D with value D. Then f belongs to Sq×q (D), where on the one hand the equality X − Mn+1 (w)

√ −1√ −1 f (w) + E I + E ∗ f (w) = −2wn+1An+1 (w) I −EE ∗ I −E ∗ E Bn+1 (w)

holds and where Step on the other hand that the function Ωf deﬁned 5 A shows + 5 by (4.1) with F := Ln+1 f Rn+1 fulﬁlls (4.4). Comparing both relations we get Ωf (w) = X. Applying Theorem 2.1 we see that Ωf belongs to Cq [D, (Γj )nj=0 ]. B A Thus X belongs to Ω(w) : Ω ∈ Cq [D, (Γj )nj=0 ] which implies ﬁnally (4.11). In view of (4.9) and (4.11) the proof is complete. Remark 4.2. Let the assumptions of Theorem 1.1 be fulﬁlled. From Theorem 1.1 and [Sm, Theorem 1.3] (see also [DFK, Theorem 1.5.2]) one can see that D C 5 5 Ω(w) : Ω ∈ Cq [D, (Γj )nj=0 ] = K Mn+1 (w); |w|n+1 2 Ln+1 (w), 2 Rn+1 (w) holds for each w ∈ D, where Mn+1 (w), Ln+1 (w), and Rn+1 (w) are given by (1.11), −1 5 −1 5 −∗ Ln+1 Iq − |w|2 Θ∗n (w)Θn (w) Ln+1 dn (w) , Ln+1 (w) := dn (w) and

−∗ 5 −1 5 −1 Rn+1 Iq − |w|2 Θn (w)Θ∗n (w) Rn+1 bn (w) . Rn+1 (w) := bn (w)

In the special case n = 0, for each w ∈ D, straightforward calculations yield that M1 (w) =

1+|w|2 1 Re Γ0 +i Im Γ0 , L1 (w) = Re Γ0 , and R1 (w) = L1 (w). 2 1−|w| 1−|w|2

Remark 4.3. Let the assumptions of Theorem 1.1 be fulﬁlled. By virtue of [FKL1, Lemma 6.1] one can get that the following statements are equivalent: (i) There is one and only one Ω ∈ Cq (D) such that (1.2) holds for each j ∈ N0,n . (ii) For each w ∈ D, the identities An+1 (w) = 0 and Bn+1(w) = 0 hold. (iii) There is some w ∈ D such that An+1 (w) = 0 or Bn+1 (w) = 0. Remark 4.4. Let the assumptions of Theorem 1.1 be fulﬁlled. Taking into account [FK1, Remark 2 in Part I], for each w ∈ D, one can immediately see that rank An+1 (w) = rank Ln+1 = rank Rn+1 = rank Bn+1 (w). Remark 4.5. Let the assumptions of Theorem 1.1 be fulﬁlled. From Remark 4.4 and [FKL1, Lemma 5.1] it follows that the following statements are equivalent: (i) (Γj )nj=0 is a nondegenerate q × q Carath´eodory sequence. (ii) For each w ∈ D, both matrices An+1 (w) and Bn+1 (w) are nonsingular. (iii) There is some w ∈ D such that An+1 (w) or Bn+1 (w) is nonsingular.

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Remark 4.6. Let the assumptions of Theorem 1.1 be fulﬁlled. Then det An+1 (w) = det Bn+1 (w) for each w ∈ D. Indeed, if (Γj )nj=0 is a nondegenerate q × q Carath´eodory sequence, then this follows from the equation det Ln+1 = det Rn+1 (see [FK1, Remark 2 in Part I]), from the identity det dn (w) = det bn (w) which holds for each w ∈ C (see [FKL1, Lemma 5.1, Remark 5.2, and Lemma 5.5] and [DGK1, equation (72)]), and from [DFK, Lemma 1.1.8]. In the degenerate case it suﬃces to apply Remark 4.5. In the rest of this section let us consider the so-called nondegenerate case which was discussed in [FK1, Part IV and Part V] in order to understand how one can guess that the Weyl matrix ball representation stated in Theorem 1.1 holds. Let n ∈ N0 . Firstly, we assume a given sequence (Γj )nj=0 of complex q × q matrices such that the matrix Γ0 is nonsingular. Then the matrix Sn deﬁned by (1.1) is nonsingular as well and there is a unique sequence (Γj )nj=0 of complex q × q matrices such that the block Toeplitz matrix ⎛ ⎞ Γ0 0 0 ... 0 ⎜ Γ Γ0 0 ... 0 ⎟ ⎜ 1 .⎟ .. ⎜ . .. ⎟ (4.12) Sn := ⎜ Γ2 Γ1 Γ0 ⎟ ⎜ .. ⎟ .. .. .. ⎝ . ⎠ . 0 . . Γn Γn−1 Γn−2 . . . Γ0 coincides with Sn−1 . This sequence (Γj )nj=0 fulﬁlls the identity Sk = Sk−1 for each k ∈ N0,n . Moreover, by setting Tn := 2 Re Sn

(4.13)

we obtain (4.14) Tn = Sn−∗Tn Sn−1 and Tn = Sn−1 Tn Sn−∗ . n Therefore, it is readily checked that (Γj )j=0 is a q×q Carath´eodory sequence if and only if (Γj )nj=0 is a q × q Carath´eodory sequence. If (Γj )nj=0 is a q × q Carath´eodory sequence with nonsingular matrix Γ0 , then (Γj )nj=0 is called the reciprocal q × q Carath´eodory sequence corresponding to (Γj )nj=0 . Now let (Γj )nj=0 be a nondegenerate q × q Carath´eodory sequence. Then the matrix Γ0 is necessarily nonsingular. Thus the matrix Sn is nonsingular and the reciprocal q × q Carath´eodory sequence (Γj )nj=0 corresponding to (Γj )nj=0 is well deﬁned. Furthermore, in view of (4.14), one can see that this q × q Carath´eodory sequence (Γj )nj=0 is also nondegenerate, i.e., the block Toeplitz matrix Tn deﬁned by (4.13) and (4.12) is positive Hermitian. Consequently, the matrix polynomials ηn , ζn , ηn , and ζn given by ηn := en,q Tn−1en,q (0),

ζn := εn,q (0)Tn−1 εn,q ,

ηn := en,q (Tn )−1 en,q (0), ζn := εn,q (0)(Tn )−1 εn,q are well deﬁned, where en,q and εn,q are the matrix polynomials given by (1.3). Moreover, the matrices Tn−1 , Ln+1 , and Rn+1 are positive Hermitian (see, e.g.,

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[FKL1, Lemma 5.1]). In view of [FK1, Theorems 28 and 29 in Part V] one can see that, if w ∈ D, then the set C D Ω(w) : Ω ∈ Cq [D, (Γj )nj=0 ] coincides with the matrix ball 5 5 K Mn+1 (w); |w|n+1 2 Ln+1 (w), 2 Rn+1 (w) , where

−1 ∗ , Ln+1 (w) := ζn∗ (w)Ln+1 ζn (w) − |w|2 η˜n[n] (w) Rn+1 η˜n[n] (w) ∗ −1 Rn+1 (w) := ηn (w)Rn+1 ηn∗ (w) − |w|2 ζ˜n[n] (w)Ln+1 ζ˜n[n] (w) ,

and

∗ [n] E [n] 2 ˜n (w) Rn+1 Γ−1 Mn+1 (w) := Ln+1 (w) ζn∗ (w)Ln+1 Γ−∗ 0 ζn (w) + |w| η 0 (ηn ) (w) . Comparing this result for the nondegenerate case with the general one stated in Theorem 1.1, for each w ∈ D, from [DFK, Corollary 1.5.1 and Theorem 1.5.2] we know that the identity Mn+1 (w) = Mn+1 (w) holds and that there is a positive real number ρn (w) such that both equalities Ln+1 (w) = ρn (w)Ln+1 (w)

and Rn+1 (w) =

1 Rn+1 (w) ρn (w)

are fulﬁlled, where Ln+1 (w) and Rn+1 (w) are the matrices deﬁned in Remark 4.2. We are going to show that actually ρn (w) = 1 holds for each w ∈ D. For this reason, ﬁrst we observe that, in the nondegenerate case we have A −1 A B B −1 Yn , Zn = Z n = Zn Tn−1 Yn = Y n = Tn−1 , and that the formulas in (4.14) and the equations −1 −1 −1 −1 ∗ ∗ Yn = Sn−1 Tn−1 , 2Zn − Zn Tn−1 Sn−1 = Zn Tn−1 Sn−1 , 2Yn − Sn−1 Tn−1

Iq Iq −1 Rn+1 Tn−1 = , −1 0 −Tn−1 Yn

−1 Iq −∗ Iq Γ0 −1 −1 Rn+1 = Rn+1 Γ 0 = Sn , Tn −1 −1 ∗ 0 Sn−1 Tn−1 Yn −Tn−1 Yn −1 −Z , 0, Iq Tn−1 = L−1 T , I n q n+1 n−1

and

−1 −1 −1 −1 −1 ∗ Γ−∗ ) = L T , I = L T S , Γ 0, I (T −Z S Z q n q n n 0 n 0 n+1 n−1 n+1 n−1 n−1

are valid. Thus we can conclude that the identities −1 , ηn = bn Rn+1

−1 ηn = an Rn+1 Γ∗0 ,

ζn = L−1 n+1 dn ,

and ζn = Γ∗0 L−1 n+1 cn (4.15)

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309

are satisﬁed (see also [FK3, Section 1]). Let w ∈ D. In view of the deﬁnition of Ln+1 (w), (3.5), and (4.15) we get −15 −15 −∗ Ln+1 (w) = dn (w) Ln+1 I − |w|2 Θ∗n (w)Θn (w) Ln+1 dn (w) −1 ∗ −1 [n] 2 ˜[n] ˜b (w) = d∗n (w)L−1 d (w) − |w| (w) R b n n n n+1 n+1 −1 ∗ = ζn∗ (w)Ln+1 ζn (w) − |w|2 η˜n[n] (w) Rn+1 η˜n[n] (w) = Ln+1 (w) and by virtue of the deﬁnition of Rn+1 (w), (3.4), and (4.15) analogously Rn+1 (w) = Rn+1(w). Note that, based on (4.15), one can also check explicitly that Mn+1(w) = Mn+1(w) is satisﬁed. Moreover, taking into account the deﬁnitions of Rn+1 (w), Ln+1 (w), ηn , and ζn as well as (1.3), (1.4), (1.5), and (1.6), an application of the Gohberg-Heinig formula (see [GH, Theorem 1.1]) for the inverse of a nonsingular block Toeplitz matrix shows again that the identities −1 1 ∗ Ln+1 (w) = εn,q (w)Tn−1 εn,q (w) 2 1−|w| and

−1 1 −1 ∗ (w)T e (w) e n,q n n,q 1−|w|2 are fulﬁlled as well (see also [FKL2, Section 6]). Rn+1 (w) =

5. On the Weyl matrix balls associated with reciprocal matrix-valued Carath´eodory sequences It is a well-known fact that if Ω ∈ Cq (D) such that det Ω(0) = 0, then det Ω(w) = 0 for each w ∈ D (see, e.g., [FK1, Remark 30 in Part V]). Taking into account this, for a ﬁxed point w ∈ D, in the following section we give a matrix ball description B A −1 : Ω ∈ Cq [D, (Γj )nj=0 ] for the case that the matrix Γ0 of the of the set Ω(w) underlying q × q Carath´eodory sequence (Γj )nj=0 is nonsingular. In fact, we present B A a similar description as given by Theorem 1.1 for Ω(w) : Ω ∈ Cq [D, (Γj )nj=0 ] , where the q × q Schur function Θn deﬁned by (3.9) is involved instead of Θn . Let n ∈ N0 and let (Γj )nj=0 be a q × q Carath´eodory sequence such that the matrix Γ0 is nonsingular. Then the matrix Sn given by (1.1) is nonsingular and hence the reciprocal q×q Carath´eodory sequence (Γj )nj=0 corresponding to (Γj )nj=0 is well deﬁned. Note that, if n ≥ 1, if Vn ∈ Yn and Wn ∈ Zn , and if we set ∗ Vn := −Sn−1 Vn Γ−1 0

then Vn =

−(Sn−1 )∗ Vn (Γ0 )−1

Vn

∗ and Wn := −Γ−1 0 Wn Sn−1 ,

and Wn = ∈

Yn

−(Γ0 )−1 Wn (Sn )∗

and

Wn

∈

Zn ,

(5.1)

as well as (5.2)

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V = Yn } and Zn := {W ∈ Cn×nq : W Tn−1 = Zn } with where Yn := {V ∈ Cnq×q : Tn−1 ⎛ ⎞ Γ1 ⎜ 1 ⎜ Γ2 ⎟ 1 ⎟ Γn , Γn−1 , . . . , Γ1 Yn := ⎜ . ⎟ and Zn := 2 ⎝ .. ⎠ 2

Γn (cf. [FK3, Lemma 3.2]). Moreover, if the matrix polynomials an , bn , cn , and dn are deﬁned as in (1.7), (1.8), (1.9), and (1.8) and if the matrix polynomials an , bn , cn , and dn are (based on (5.1)) given by 2 Γ0 if n = 0 (5.3) an (z) := )∗ Vn if n ≥ 1 , Γ0 + zen−1,q (z)(Sn−1 2 Iq if n = 0 bn (z) := (5.4) Iq − zen−1,q (z)Vn if n ≥ 1 , 2 Γ0 if n = 0 (5.5) cn (z) := ∗ Wn (Sn−1 ) zεn−1,q (z) + Γ0 if n ≥ 1 , and 2 Iq if n = 0 dn (z) := (5.6) −Wn zεn−1,q (z) + I if n ≥ 1 with some Vn ∈ Yn and Wn ∈ Zn in the case of n ≥ 1, then it is readily checked that the equalities an = bn Γ−1 0 ,

bn = an Γ−1 0 ,

cn = Γ−1 0 dn ,

and dn = Γ−1 0 cn

(5.7)

hold. Using these identities, we obtain the following interrelation between the construction of q × q Schur functions stated in Propositions 3.4 and 3.8. Proposition 5.1. Let n ∈ N0 , let (Γj )nj=0 be a q ×q Carath´eodory sequence such that the matrix Γ0 is nonsingular, and let (Γj )nj=0 be the reciprocal q × q Carath´eodory sequence corresponding to (Γj )nj=0 . If n ≥ 1, then let Vn ∈ Y n , let Wn ∈ Z n , let −1 ∗ ∗ Vn := −Sn−1 Vn Γ−1 0 , and let Wn := −Γ0 Wn Sn−1 . Furthermore, let the matrix polynomials an , bn , cn , dn , an , bn , cn , and dn be deﬁned by (1.7), (1.8), (1.9), (1.10), (5.3), (5.4), (5.5), and (5.6) and let the matrix-valued functions Θn , Θn , Θn , and Ξn be given by (3.4), (3.9), 7 + [n] 7 6 (bn )−1 (dn ) Ln+1 , (5.8) Θn := Rn+1 and

7 Ξn :=

6 [n] Rn+1 (an )−1 (cn )

7 + Ln+1 ,

where L1 := Re Γ0 ,

R1 := Re Γ0

(5.9)

in the case n = 0 and Ln+1 := Re Γ0 − Zn (Tn−1 )+ (Zn )∗ ,

Rn+1 := Re Γ0 − (Yn )∗ (Tn−1 )+ Yn

(5.10)

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311

if n ≥ 1. There are unitary q × q matrices U and V fulﬁlling 7 7 5 5 −1 ∗ ∗ L = L U and R Γ = V Rn+1 Γ−1 n+1 n+1 n+1 0 0

(5.11)

and, if U and V are such unitary q × q matrices, then Θn = Rn+1 (Rn+1 )+ V Θn U,

Θn = V Θn U Ln+1 (Ln+1 )+

(5.12)

(Rn+1 )+ V Θn U, Ξn = Rn+1

Ξn = V Θn U Ln+1 (Ln+1 )+ .

(5.13)

and In particular, if Γ0 = I, then

Θn

=

Θn

and Ξn = Θn .

Proof. Using [FK3, the formulas by (3.15)] and the polar decomposition of matrices we get that there are unitary q ×q matrices U and V such that the identities stated in (5.11) are satisﬁed. Lemma 3.2 yields ˜[n] c−1 Ln+1 . (5.14) Rn+1 b−1 d˜[n] = ˜b[n] d−1 Ln+1 and Rn+1 a−1 c˜[n] = a n

n

n

n

n

n

n

n

Because of (5.8), (5.7), (5.14), (5.11), and (3.10) we can conclude 7 7 + + [n] 7 5 −1 6 −1 [n] −∗ Θn = (d Rn+1 (bn ) Ln+1 = V Rn+1 an c˜n Γ0 Ln+1 n) 7 7 + + 5 5 + [n] −1 = V Rn+1 a ˜n cn Ln+1 Γ−∗ Ln+1 = V Θn Ln+1 Γ−∗ Ln+1 0 0 7 ∗ 7 + = V Θn Ln+1 U ∗ Ln+1 = V Θn U Ln+1 (Ln+1 )+ . Thus the second equation in (5.12) is checked. Similarly, based on (5.7), (5.14), and (5.11) the ﬁrst equation in (5.12) and the identities in (5.13) can be proved. In the particular case Γ0 = Iq one can choose U = V = Iq (note [FK3, the formulas in (3.15)] and (5.11)) so that Θn = Θn and Ξn = Θn follow from (5.7). Theorem 5.2. Let n ∈ N0 and let (Γj )nj=0 be a q × q Carath´eodory sequence such that the matrix Γ0 is nonsingular. If n ≥ 1, then let Vn ∈ Y n and let Wn ∈ Z n . Let the matrix polynomials an , bn , cn , and dn be deﬁned by (1.7), (1.8), (1.9), and (1.10). Let the matrices Ln+1 , Rn+1 , Ln+1 , and Rn+1 be given by (1.4), n (1.6), (5.9), and (5.10), where (Γj )j=0 stands for the reciprocal q × q Carath´eodory sequence corresponding to (Γj )nj=0 . If Ω ∈ Cq [D, (Γj )nj=0 ], then the function det Ω does not vanish in D and D C −1 : Ω ∈ Cq [D, (Γj )nj=0 ] = K Mn+1 (w); 2|w|n+1 An+1 (w), Bn+1 (w) Ω(w) for each w ∈ D with −1 [n] 2 ˜b[n] (w) , (5.15) (w)˜ a (w) (w) + |w| G (w) d Mn+1 (w) := cn (w) − |w|2 G n n n n n 7 −1 −1 5 ∗ An+1 (w) := cn (w) Ln+1 Iq − |w|2 Θn (w) Θn (w) , and Bn+1 (w) :=

7

∗ −1 5 −1 Iq − |w|2 Θn (w) Θn (w) Rn+1 an (w) ,

(5.16)

(5.17)

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5 ∗ 5 + Ln+1 Θn (w) Rn+1 Γ∗0 Rn+1 (Rn+1 )+ Γ−∗ and where Θn where G n (w) := 0 stands for the rational matrix-valued function deﬁned by (3.9). Moreover, for each w ∈ D, the matrix Mn+1 (w) can be represented via −1 , Mn+1 (w) = bn (w) + |w|2 d˜n[n] (w)Fn (w) an (w) − |w|2 c˜[n] n (w)Fn (w) + ∗ where Fn (w) := Γ−∗ 0 Ln+1 (Ln+1 ) Γ0

5

Ln+1

∗ 5 Rn+1 . Θn (w)

+

Proof. For each Ω ∈ Cq [D, (Γj )nj=0 ] we have det Ω(0) = det Γ0 = 0. Consequently, the inequality det Ω(w) = 0 holds for each w ∈ D and D C (5.18) Ω−1 : Ω ∈ Cq [D, (Γj )nj=0 ] = Cq [D, (Γj )nj=0 ] is valid (see, e.g., [FK1, Remark 30 in Part V]). Let the matrix polynomials an , bn , cn , and dn be deﬁned by (5.3), (5.4), (5.5), and (5.6) based on (5.1). Let w ∈ D. Due to (5.18), (5.2), (5.7), and Remark 3.7 an application of Theorem 1.1 yields C D −1 Ω(w) : Ω ∈ Cq [D, (Γj )nj=0 ] = K Mn+1 (w); 2|w|n+1 An+1 (w), Bn+1 (w) with the settings −1 [n] [n] 6 6 Mn+1 (w) := dn (w) − |w|2 G•n (w)(bn ) (w) cn (w) + |w|2 G•n (w)(an ) (w) , −1 −1 7 7 ∗ Ln+1 I − |w|2 Θn (w) Θn (w) , An+1 (w) := dn (w)

and

7 ∗ −1 7 −1 Bn+1 (w) := I − |w|2 Θn (w) Θn (w) Rn+1 bn (w) , 7 7 + ∗ Rn+1 and where Θn is the matrix-valued where G•n (w) := Ln+1 Θn (w) function deﬁned by (5.8). From Theorem 1.1 we also see that −1 [n] [n] 6 6 Mn+1 (w) = an (w) + |w|2(cn ) (w)Fn• (w) bn (w) − |w|2(dn ) (w)Fn• (w) (5.19)

7 + ∗ 7 Rn+1 . According to Proposition 5.1 holds, where Fn• (w) := Ln+1 Θn (w) there are unitary q × q matrices U and V such that (5.11), (5.12), and (5.13) are valid. Using (5.11), (5.12), and (3.9) we obtain ∗ 7 + 5 + L U R (R ) V Θ (w)U R G•n (w) = Γ−1 n+1 n n+1 n+1 n+1 0 7 + 5 ∗ = Γ−1 Ln+1 Θn (w) V ∗ Rn+1 0 5 5 5 +5 + ∗ = Γ−1 Ln+1 Θn (w) Rn+1 Rn+1 V ∗ Rn+1 0 7 7 + 5 ∗ 5 + ∗ Θ = Γ−1 L (w) R Γ R R n+1 n+1 n 0 n+1 n+1 0 5 ∗ 5 + ∗ + Θ = Γ−1 L (w) R Γ R (R ) (5.20) n+1 n+1 n 0 n+1 0 n+1

Weyl Matrix Balls and analogously Fn• (w) = Ln+1 (Ln+1 )+ Γ∗0

5 ∗ 5 + Ln+1 Θn (w) Rn+1 Γ−1 0 .

313

(5.21)

In addition, from (5.12), (3.10), and (5.11) we also get ∗ Θn (w) Θn (w) ∗ = U ∗ Θn (w) V ∗ (Rn+1 )+ Rn+1 V Θn (w)U 5 5 5 ∗5 + + ∗ Rn+1 Rn+1 V ∗ (Rn+1 )+ Rn+1 Rn+1 V ∗ Rn+1 Θn (w)U = U ∗ Θn (w) 7 7 5 ∗ 5 + + = U ∗ Θn (w) Rn+1 Γ∗0 Rn+1 Rn+1 Γ0 Rn+1 Θn (w)U 5 ∗ 5 ∗ + + = U ∗ Θn (w) Rn+1 Rn+1 Rn+1 Θn (w)U = U ∗ Θn (w) Θn(w)U (5.22) and analogously

∗ ∗ Θn (w) Θn (w) = V Θn (w) Θn (w) V ∗ .

(5.23)

Because of (5.7) and (5.20) we have 6 [n] dn (w) − |w|2 G•n (w)(bn ) (w) = Γ−1 a[n] cn (w) − |w|2 G n (w)˜ n (w) 0 and

[n] 6 2 ˜b[n] (w) . cn (w) + |w|2 G•n (w)(an ) (w) = Γ−1 (w) + |w| G (w) d n n n 0

Thus we see that Mn+1 (w) is a well-deﬁned matrix and that the identity Mn+1 (w) = Mn+1 (w)

(5.24)

holds. Furthermore, from (5.19), (5.7), and (5.21) it follows the second representation of Mn+1 (w). By virtue of (5.7), (5.11), and (5.22) we can conclude 7 −1 −1 5 ∗ An+1 (w) = cn (w) Ln+1U I − |w|2 U ∗ Θn(w) Θn (w)U 7 −1 −1 5 ∗ = cn (w) Ln+1 I − |w|2 Θn (w) Θn (w) U = An+1 (w)U and because of (5.23) analogously (w) = V Bn+1 (w). Bn+1

Application of [Sm, Theorem 1.3] completes the proof.

Remark 5.3. Let the assumptions of Theorem 5.2 be fulﬁlled. Then Theorem 5.2 and [Sm, Theorem 1.3] (see also [DFK, Theorem 1.5.2]) imply the identity 7 7 C D −1 Ω(w) : Ω ∈ Cq [D, (Γj )nj=0 ] = K Mn+1(w); |w|n+1 2Ln+1(w), 2Rn+1(w) for each w ∈ D, where Mn+1 (w), Ln+1 (w), and Rn+1 (w) are given by (5.15), −1 5 −1 5 ∗ −∗ Ln+1 (w) := cn (w) Ln+1 Iq − |w|2 Θn (w) Θn (w) Ln+1 cn (w) ,

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and

−∗ 5 ∗ −1 5 −1 Rn+1 Iq − |w|2 Θn(w) Θn (w) Rn+1 an (w) . Rn+1 (w) := an (w)

Remark 5.4. Let the assumptions of Theorem 5.2 be fulﬁlled and let Γ0 = Iq . Then (5.11) shows that Ln+1 = Ln+1 and Rn+1 = Rn+1 hold. Thus it is readily checked that, for each w ∈ D, the matrices G n (w) and Fn (w), which lead to representations of Mn+1 (w) according to Theorem 5.2, have the simpler form 5 5 ∗ 5 ∗ 5 + + G Ln+1 Θn (w) Rn+1 and Fn (w) = Ln+1 Θn (w) Rn+1 n (w) = in that case. (These formulas are already satisﬁed if Γ0 is a unitary q × q matrix.) Remark 5.5. Let the assumptions of Theorem 5.2 be fulﬁlled. In view of [FK1, Remark 2 in Part I], (5.11), and Theorem 5.2 one can see that rank An+1 (w) = rank Ln+1 = rank Rn+1 = rank Bn+1 (w)

for every choice of w in D. Let the assumptions of Theorem 5.2 be fulﬁlled. Based on Remarks 4.4, 5.5, and 4.3 one can immediately get further necessary and suﬃcient conditions for the fact that the solution set Cq [D, (Γj )nj=0 ] of the matricial Carath´eodory problem contains exactly one element. Moreover, application of Remarks 4.4, 5.5, and 4.5 yields further necessary and and suﬃcient conditions for the nondegeneracy of a given q × q Carath´eodory sequence (Γj )nj=0 . We omit the details. Remark 5.6. Let the assumptions of Theorem 5.2 be fulﬁlled. From Remark 4.6 and (5.7) one can see that the identity det an (w) = det cn (w) holds for each w ∈ D. Thus taking into account [FK1, Remark 2 in Part I], [DFK, Lemma 1.1.8], (5.16), and (5.17) for each w ∈ D it follows (w). det An+1 (w) = det Bn+1

6. Further observations on the matrix-valued functions Θn and Θn Theorem 1.1 shows that the rational matrix-valued function Θn deﬁned by (3.4) plays a key role in the description of the Weyl matrix balls of the solutions of the matricial Carath´eodory problem. In this section, we check that this function Θn (and hence the parameters of the Weyl matrix balls deﬁned by Theorem 1.1) does not depend on the concrete choice of the underlying matrix Vn in the set Y n and the underlying matrix Wn in the set Z n . Furthermore, we will state a possibility to construct recursively the function Θn . Since there are matrix polynomials an , bn , cn , and dn of the types which are considered in Theorem 1.1 which can be constructed recursively as well (see [FK3, Proposition 4.4, Remark 4.5, and Lemma 4.6]) one gets a particular possibility to calculate the parameters of the Weyl matrix balls of the solutions of matricial Carath´eodory problem. Moreover, we

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present analogous results corresponding to the rational matrix-valued function Θn deﬁned by (3.9), which appears in the Weyl matrix balls studied in Theorem 5.2. At ﬁrst, we show that the functions Θn and Θn are independent of the concrete choice of Vn in Y n and Wn in Z n . In other words, the functions Θn and Θn depend only on the given q × q Carath´eodory sequence (Γj )nj=0 . Proposition 6.1. Let n ∈ N0 and let (Γj )nj=0 be a q × q Carath´eodory sequence. If n ≥ 1, then let Vn ,Vn ∈ Y n and let Wn ,Wn ∈ Z n . Let an , bn , cn , and dn be the matrix polynomials given by (1.7), (1.8), (1.9), and (1.10), and let an , bn , cn , and dn be the matrix polynomials which are deﬁned analogously to an , bn , cn , and dn using (if n ≥ 1) the matrices Vn and Wn instead of Vn and Wn , respectively. Furthermore, let the matrices Ln+1 and Rn+1 be given by (1.4) and (1.6). Then: (a) The matrix-valued functions 5 5 ˜[n] Ln+1 + Θn := Rn+1 b−1 n dn

and

Θn :=

5 5 ˜ [n] Ln+1 + Rn+1 b−1 n dn

coincide. (b) If the matrix Γ0 is nonsingular, then the matrix-valued functions 5 5 5 5 + + ˜[n] Ln+1 and Θn := Rn+1 a−1 Ln+1 Θn := Rn+1 a−1 c[n] n c n n ˜ n coincide. Proof. (a) Let w ∈ D \ {0}. Using Theorem 1.1 in combination with [DFK, Corollary 1.5.1] we can conclude −1 2 ˜[n] (w)F (w) b (w) − |w| (w)F (w) an (w) + |w|2 c˜[n] d n n n n n −1 2 [n] 2 ˜ [n] = an (w) + |w| ˜ , (6.1) cn (w)Fn (w) bn (w) − |w| dn (w)Fn (w) 5 5 5 5 + + where Fn (w) := Ln+1 Θ∗n (w) Rn+1 and Fn (w) := Ln+1 Θ∗n (w) Rn+1 . According to Proposition 3.4 the matrix −wΘ∗n (w) is strictly contractive. Thus [FKL1, Lemma 3.1] yields −1 2 ˜ [n] (w)F (w) b (w) − |w| (w)F (w) c[n] an (w) + |w|2 ˜ d n n n n n −1 = an (w) + |w|2 c˜[n] bn (w) − |w|2 d˜n[n] (w)Fn (w) . n (w)Fn (w) Combining this with (6.1) we obtain −1 an (w) + |w|2 c˜[n] bn (w) − |w|2 d˜n[n] (w)Fn (w) n (w)Fn (w) −1 = an (w) + |w|2 c˜[n] bn (w) − |w|2 d˜n[n] (w)Fn (w) . n (w)Fn (w) Since we see from Proposition 3.4 that the matrices −wΘ∗n (w) and −wΘ∗n (w) are both contractive, Proposition 2.2 provides us + + + ∗ ∗ −wLn+1 L+ n+1 Θn (w)Rn+1 Rn+1 = −wLn+1 Ln+1 Θn (w)Rn+1 Rn+1 .

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Therefore, in view of w = 0 and the deﬁnitions of Θn and Θn , we get the identity Θ∗n (w) = Θ∗n (w), i.e., Θn (w) = Θn (w). Since Θn and Θn are rational matrixvalued functions, we can ﬁnally conclude Θn = Θn . (b) Based on the reciprocal q × q Carath´eodory sequence (Γj )nj=0 to (Γj )nj=0 , the assertion of part (b) follows from (a) in combination with Proposition 5.1. Corollary 6.2. Let the assumptions of Proposition 6.1 be fulﬁlled. Then: 5 5 5 5 (a) The identities d−1 Ln+1 = d−1 Ln+1 and Rn+1 b−1 Rn+1 b−1 n n n5= n5 hold. −1 −1 Ln+1 = cn Ln+1 (b) If the matrix Γ0 is nonsingular, then the equalities cn 5 5 −1 and Rn+1 a−1 = R a are satisﬁed. n+1 n n Proof. First we observe that one can easily see that each of the sets Nbn , N˜b[n] , n Ndn , Nd˜[n] , Nbn , Nb˜ [n] , Ndn , and Nd˜ [n] consists of at most nq elements. Taking n n n into account Lemma 3.2 we obtain ˜ [n] = ˜b[n] d−1 Ln+1 . Rn+1 b−1 d˜[n] = ˜b[n] d−1 Ln+1 and Rn+1 b−1 d (6.2) n

n

n

n

n

n

n

n

Moreover, according to part (a) of Proposition 6.1, the rational matrix functions 5 5 ˜ [n] Ln+1 + coincide. Thus from (6.2) it follows Θn and Rn+1 b−1 n dn 5 5 5 5 ˜b[n] d−1 Ln+1 = ˜b[n] d−1 Ln+1 Ln+1 + = Rn+1 b−1 d˜[n] Ln+1 + = Rn+1 Θn n n n n n n 5 5 5 ˜ [n] Ln+1 + = ˜b[n] d−1 Ln+1 Ln+1 + = ˜b[n] d−1 Ln+1 = Rn+1 b−1 n dn n n n n 5 5 ˜[n] ˜[n] and similarly Rn+1 b−1 Rn+1 b−1 n dn = n dn . Using continuity arguments we get 5 5 5 5 Ln+1 = d−1 Ln+1 and Rn+1 b−1 Rn+1 b−1 d−1 n n n = n . Hence part (a) is proved. Applying Lemma 3.2, part (b) of Proposition 6.1, and Proposition 3.8, part (b) can be veriﬁed analogously. Remark 6.3. Let the assumptions of Proposition 6.1 be fulﬁlled. Further, suppose that the matrix Γ0 is nonsingular. Because of Corollary 6.2 one can immediately conclude that the matrix-valued function Θn given by (3.17) coincides with 5 5 −1 Ln+1 . Θn := Rn+1 a−1 n dn Based on part (a) of Proposition 6.1 we get that, for each w ∈ D, the semiradii of the matrix ball described in Theorem 1.1 is independent of the concrete choice of the matrix Vn in Y n and the matrix Wn in Z n . Remark 6.4. Let the assumptions of Proposition 6.1 be fulﬁlled. From Proposition 6.1 and Corollary 6.2 one can see that the identities An+1 (w) = An+1 (w) and Bn+1 (w) = Bn+1 (w) hold for each w ∈ D, where An+1 (w) and Bn+1 (w) are deﬁned by (1.12) and (1.13) and where similarly 5 −1 5 −1 An+1 (w) := dn (w) Ln+1 I − |w|2 Θ∗n (w)Θn (w) and Bn+1 (w) :=

5 −1 −1 5 I − |w|2 Θn (w)Θ∗n (w) Rn+1 bn (w) .

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The semi-radii of the matrix ball described in Theorem 5.2 do not depend on the concrete choice of Vn in Y n and Wn in Z n as well. Remark 6.5. Let the assumptions of Proposition 6.1 be fulﬁlled. Further, suppose that the matrix Γ0 is nonsingular. From Proposition 6.1 and Corollary 6.2 it follows that An+1 (w) = An+1 (w) and Bn+1 (w) = Bn+1 (w) hold for each w ∈ D, where An+1 (w) and Bn+1 (w) are deﬁned by (5.16) and (5.17) and where similarly 7 −1 −1 5 ∗ An+1 (w) := cn (w) Ln+1 I − |w|2 Θn (w) Θn (w) and Bn+1 (w) :=

7 ∗ −1 5 −1 I − |w|2 Θn (w) Θn (w) Rn+1 an (w) .

Now we are going to verify recurrence relations for the rational matrix-valued functions Θn and Θn deﬁned by (3.4) and (3.9), respectively. Before, we state some technical results which are useful in view of the proof of these formulas. Remark 6.6. Let A ∈ Cp×q and let B ∈ Cq×p . Then det(I − BA) = det(I − AB) holds (see, e.g., [DFK, Lemma 1.1.8]). Moreover, if det(I − AB) = 0, then the identity (I − AB)−1 A = A(I − BA)−1 is fulﬁlled. Remark 6.7. Let n ∈ N and let (Γj )nj=0 be a q × q Carath´eodory sequence. Let a0 , b0 , c0 , and d0 be the constant matrix-valued functions given, for each z ∈ C, by a0 (z) := Γ0 ,

b0 (z) := Iq ,

c0 (z) := Γ0 ,

d0 (z) := Iq ,

and for each m ∈ N0,n−1 let the matrix polynomials am+1 , bm+1 , cm+1 , and dm+1 be recursively deﬁned, for each z ∈ C, by ˜ [m] (z)tm+1 , am+1 (z) := am (z) + z˜ c[m] (z)tm+1 , bm+1 (z) := bm (z) − z d m

m

˜[m] zum+1 a m (z),

cm+1 (z) := cm (z) + where 1 tm+1 := L+ m+1 2 Γm+1 − Mm+1

˜ [m] (z), dm+1 (z) := dm (z) − zum+1 b m

and um+1 :=

1

2 Γm+1

+ − Mm+1 Rm+1 . (6.3)

In view of [FK3, Remark 4.5, Proposition 4.4, and Lemma 4.6] one can see that, for each m ∈ N1,n , there are matrices Vm ∈ Y m and Wm ∈ Z m such that ∗ am (z) = Γ0 + zem−1,q (z)Sm−1 Vm ,

bm (z) = Iq − zem−1,q (z)Vm ,

∗ Wm Sm−1 zεm−1,q (z)

+ Γ0 , dm (z) = −Wm zεm−1,q (z) + Iq cm (z) = hold. In particular, for each m ∈ N0,n and each w ∈ D, the complex q × q matrices bm (w) and dm (w) are nonsingular. If (Γj )nj=0 is a q × q Carath´eodory sequence, then we will use the notation = if k = 0 0q×q Mk+1 := (6.4) + Zk Tk−1 Yk if k ∈ N1,n , where for each k ∈ N1,n the matrices Zk and Yk are given by (1.5). Furthermore, let Θn be the matrix-valued function deﬁned by (3.4).

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Theorem 6.8. Let n ∈ N0 and let (Γj )nj=0 be a q × q Carath´eodory sequence. Then: (a) Θ0 is the constant function with value (Re Γ0 )(Re Γ0 )+ . (b) If n ≥ 1, then for each m ∈ N1,n the matrix 5 5 + + 1 Γ − Mm Rm Km := Lm 2 m

(6.5)

is contractive and the recurrence formulas 5 5 −15 +5 + ∗ Lm Lm+1 Θm (w) = Rm+1 Rm wΘm−1 (w) − Km I − wKm Θm−1 (w) and

5 + 5 −1 5 5 + ∗ Lm Lm+1 wΘm−1 (w) − Km Θm (w) = Rm+1 Rm I − wΘm−1 (w)Km hold for every choice of w in D. Proof. Since (Γj )nj=0 is a q × q Carath´eodory sequence, for each m ∈ N0,n the sequence (Γj )m eodory sequence as well. According to Proposij=0 is a q × q Carath´ tion 3.4, for each m ∈ N0,n the rational matrix-valued function Θm is well deﬁned and its restriction onto D belongs to Sq×q (D). Moreover, part (a) follows immediately from (1.4), (1.8), (1.10), and a well-known statement on Moore-Penrose inverses (see, e.g., [DFK, Lemma 1.1.6]). We consider now the case n ≥ 1. For each m ∈ N0,n , let the matrix polynomials am , bm , cm , and dm be deﬁned as in Remark 6.7. For each m ∈ N1,n , from [FKL1, Remark 2.1] we get that the matrix Km is contractive and consequently that for each w ∈ D the matrices I − wKm Θm−1 (w) and I − wΘm−1 (w)Km are nonsingular. From Remark 6.7, Proposition 3.4, and Proposition 6.1 we can conclude that, for each j ∈ N0,n and each w ∈ D, the rational matrix-valued function Θj can be represented via 5 5 + [j] ˜ (w) dj (w) −1 Lj+1 (6.6) Θj (w) = Rj+1 b j and Θj (w) =

5

5 −1 [j] ˜ (w) Lj+1 + . Rj+1 bj (w) d j

(6.7)

Now let m ∈ N1,n . Using Remark 6.7 we get 5 + ˜ [m−1] (w) − t∗ dm−1 (w) Θm (w) = Rm+1 wb m m−1 5 ˜ [m−1] (w) −1 Lm+1 (w) − wum b · dm−1 m−1 5 + ˜ [m−1] (w) dm−1 (w) −1 − t∗ = Rm+1 wb m m−1 −1 5 [m−1] ˜ · I − wum b Lm+1 m−1 (w) dm−1 (w) for each w ∈ D. Because of (6.5), (6.3), and [FKL1, Remark 2.1] we have 5 5 5 5 + + tm = Lm Km Rm and um = Lm Km Rm . Furthermore, the identities 5 5 5 +5 Lm L m Lm+1 = Lm+1

and

5 5 5 + +5 + Rm+1 Rm Rm = Rm+1

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hold (see [DFK, Remark 3.4.3]). Thus it follows 5 5 5 −1 + [m−1] + +5 ∗ ˜ Θm (w) = Rm+1 Rm w Rm b − Km Lm m−1 (w) dm−1 (w) 5 5 5 −1 −15 5 + [m−1] + ˜ · I −w Lm Km Rm b Lm Lm Lm+1 m−1 (w) dm−1 (w) and consequently by virtue of Remark 6.6 then 5 5 5 −1 + [m−1] + +5 ∗ ˜ Θm (w) = Rm+1 Rm w Rm b − Km Lm m−1 (w) dm−1 (w) 5 5 5 −15 −15 + [m−1] + ˜ · Lm I −wKm Rm b Lm Lm Lm+1 m−1 (w) dm−1 (w) √ +√ ∗ ∗ for each w ∈ D. Thus, in view of (6.6) and Km Lm Lm = K m we obtain that 5 5 5 5 −1 + + ∗ I − wKm Θm−1 (w) Lm Lm+1 Θm (w) = Rm+1 Rm wΘm−1 (w) − Km holds for each w ∈ D. Using (6.7), the other recurrence formula of part (b) can be proved analogously. Corollary 6.9. Let n ∈ N and let (Γj )nj=0 be a q × q Carath´eodory sequence. For each m ∈ N1,n , then 5 5 +5 +5 ∗ Rm Km Lm Lm+1 Θm (0) = − Rm+1 and

5 5 5 5 + + ∗ Θm (0) = − Rm+1 Rm Km Lm Lm+1 ,

where Km is the matrix given by (6.5). Moreover, if m ∈ N1,n and if w ∈ D are such that Θm−1 (w) = 0, then Θm (w) = Θm (0).

Proof. Apply Theorem 6.8.

Corollary 6.10. Let n ∈ N and let (Γj )nj=0 be a q × q Carath´eodory sequence which satisﬁes Γn = 2Mn , where Mn is deﬁned by (6.4). For each w ∈ D, then Θn (w) = wΘn−1 (w). Proof. In view of Γn = 2Mn , an application of [DFK, Remark 3.4.3] implies Ln+1 = Ln

and Rn+1 = Rn

as well as (6.5) yields Kn = 0q×q . Thus part (b) of Theorem 6.8 and (3.4) lead to 5 +5 5 +5 Θn (w) = Rn+1 Rn wΘn−1 (w) Ln Ln+1 = wΘn−1 (w) for each w ∈ D.

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Let n ∈ N and let (Γj )nj=0 be a q×q Carath´eodory sequence. Since the matrixvalued functions Θ0 , Θ1 , . . . , Θn given by (3.4) are rational, there is a ﬁnite subset N of C \ D such that the recurrence formulas stated in part (b) of Theorem 6.8 hold actually for each m ∈ N1,n and each w ∈ C \ N . Now we are going to show that if a q × q Carath´eodory sequence with nonsingular matrix Γ0 is given, then the matrix-valued functions Θ0 , Θ1 , . . . , Θn deﬁned by (3.9) fulﬁll some recurrence relations as well. Theorem 6.11. Let n ∈ N and let (Γj )nj=0 be a q × q Carath´eodory sequence such that the matrix Γ0 is nonsingular. Then: √ √ + Re Γ0 . (a) Θ0 is the constant function with value Re Γ0 Γ∗0 Γ−1 0 (b) If n ≥ 1, then for each m ∈ N1,n and each w ∈ D the recurrence formulas 5 −15 +5 +5 ∗ Lm Lm+1 Θm (w) = Rm+1 Rm wΘm−1 (w) + Km I + wKm Θm−1 (w) and

5 + 5 −1 5 5 + ∗ wΘm−1 (w) + Km Θm (w) = Rm+1 Rm I + wΘm−1 (w)Km Lm Lm+1 hold, where Km is the contractive matrix given by (6.5). Proof. Using Remark 3.7, Proposition 3.8, part (b) of Proposition 6.1, Remark 6.6, and Remark 6.7 one can prove Theorem 6.11 analogously to Theorem 6.8. We omit the details. Corollary 6.12. Let n ∈ N and let (Γj )nj=0 be a q × q Carath´eodory sequence with nonsingular matrix Γ0 . For each m ∈ N1,n , then Θm (0) = −Θm (0). Moreover, if m ∈ N1,n and w ∈ D are such that Θm−1 (w) = 0, then Θm (w) = Θm (0). Proof. Use Theorem 6.11 and Corollary 6.9.

Corollary 6.13. Let n ∈ N and let (Γj )nj=0 be a q × q Carath´eodory sequence such that the matrix Γ0 is nonsingular and that the identity Γn = 2Mn is fulﬁlled, where Mn is deﬁned by (6.4). Then Θn (w) = wΘn−1 (w) for each w ∈ D. Proof. Using the same argumentation as in the proof of Corollary 6.10, the assertion is an easy consequence of part (b) of Theorem 6.11. Let n ∈ N and let (Γj )nj=0 be a q × q Carath´eodory sequence with nonsingular matrix Γ0 . Since Θ0 , Θ1 , . . . , Θn are rational matrix-valued functions, there is a ﬁnite subset M of C \ D such that the recurrence formulas in Theorem 6.11 hold actually for each m ∈ N1,n and each w ∈ C \ M. Furthermore, we note marginally that one can also verify recurrence relations for the rational matrixvalued functions Θ0 , Θ1 , . . . , Θn given by (3.17) using the same argumentation as in the proof of Theorem 6.8 (respectively, Theorem 6.11), where the function Θm can be calculated based on Θm−1 , Θm−1 , Θm−1 , and Km for each m ∈ N1,n .

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7. Some remarks on central q × q Carath´eodory functions In the present section, we are going to describe the situation that, starting from a q × q Carath´eodory sequence (Γj )nj=0 and a point w belonging to the unit disk D, the value Ωc,n (w) of the central q × q Carath´eodory function Ωc,n corresponding to (Γj )nj=0 coincides with the center Mn+1 (w) of the Weyl matrix ball which is B A given by the value set Ω(w) : Ω ∈ Cq [D, (Γj )nj=0 ] according to Theorem 1.1. Let us consider an arbitrary nonnegative integer n and an arbitrary q × q Carath´eodory sequence (Γj )nj=0 . Then, in view of [DFK, Theorem 3.4.1], the set of all complex q ×q matrices Γn+1 such that (Γj )n+1 is a q ×q Carath´eodory sequence 5 j=0 5 coincides with the matrix ball K 2Mn+1 ; 2Ln+1 , 2Rn+1 , where the matrices Mn+1 , Ln+1 , and Rn+1 are deﬁned by (6.4), (1.4), and (1.6). Thus the choice Γn+1 := 2Mn+1 yields a particular q × q Carath´eodory sequence (Γj )n+1 j=0 and the is a q × q Carath´ eodory set of all complex q × q matrices Γn+2 such that (Γj )n+2 j=0 5 5 sequence coincides with K 2Mn+2 ; 2Ln+2 , 2Rn+2 . In this way, choosing Γn+1+k := 2Mn+1+k

(7.1) (Γj )∞ j=0

for each k ∈ N0 one obtains a particular q × q Carath´eodory sequence and hence a particular function which belongs to Cq [D, (Γj )nj=0 ], the so-called central q × q Carath´eodory function Ωc,n corresponding to (Γj )nj=0 . Clearly, it admits the Taylor series representation ∞ Ωc,n(z) = Γk z k , z ∈ D, k=0

where for each k ∈ N0 the matrices Γn+1+k are deﬁned by (7.1). Moreover, for each z ∈ D, it admits the representations −1 −1 and Ωc,n (z) = dn (z) cn (z), (7.2) Ωc,n (z) = an (z) bn (z) where Vn and Wn are arbitrary matrices which belong to Y n and Z n , respectively, and where an , bn , cn , and dn are the matrix polynomials deﬁned by (1.7), (1.8), (1.9), and (1.10) (see [FK3, Remark 1.1 and Theorems 1.2., 1.7, and 2.3]). In the following, we use again the notations of Theorem 1.1 and Remark 4.2. Theorem 7.1. Let n ∈ N0 , let (Γj )nj=0 be a q × q Carath´eodory sequence, let Ωc,n be the central q × q Carath´eodory function corresponding to (Γj )nj=0 , and let w ∈ D. Furthermore, let the matrices Ln+1 and Rn+1 be deﬁned by (1.4) and (1.6). Then the following statements are equivalent: (i) Ωc,n (w) = Mn+1 (w). (ii) wΘn (w) = 0. −1 5 Ln+1 . (iii) An+1 (w) = dn (w) 5 −1 . (iv) Bn+1 (w) = Rn+1 bn (w) −1 −∗ Ln+1 dn (w) . (v) Ln+1 (w) = dn (w) −∗ −1 Rn+1 bn (w) . (vi) Rn+1 (w) = bn (w)

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Proof. (i) ⇔ (ii): If w = 0, the equivalence of (i) and (ii) is an easy consequence of (7.2) and∗(1.11). Now suppose w = 0. In view of Proposition 3.4 we see that −wΘn (w) is contractive. Because of (7.2) we have on the one hand Ωc,n (w) = an (w)(bn (w))−1 −1 = −w˜ c[n] wd˜[n] n (w)F1 (w) + an (w) n (w)F1 (w) + bn (w) 5 + 5 with F1 (w) := Ln+1 0 Rn+1 , where on the other hand (1.14) implies −1 ˜[n] (w)F2 (w) + bn (w) Mn+1 (w) = −w˜ c[n] (w)F (w) + a (w) w d 2 n n n 5 ∗ 5 + with F2 (w) := Ln+1 −wΘn (w) Rn+1 . Therefore, in view of w = 0 and part (a) of Proposition 2.2 we get that (i) is necessary and suﬃcient for ∗ + + + Ln+1 L+ n+1 −wΘn (w) Rn+1 Rn+1 = Ln+1 Ln+1 0Rn+1 Rn+1 , i.e., (i) is equivalent to

+ Rn+1 Rn+1 wΘn (w) Ln+1 L+ n+1 = 0.

Thus, because of (3.4), statement (i) is also tantamount to (ii) in this case. (ii) ⇒ (iii): This implication follows immediately from (1.12). (iii) ⇒ (v): Since the deﬁnitions of An+1 (w) and Ln+1 (w) imply the relation An+1 (w)A∗n+1 (w) = Ln+1 (w), we can directly see that (iii) leads to (v). (v) ⇒ (ii): Because of (v) we obtain −1 5 −1 5 −∗ dn (w) Ln+1 I − |w|2 Θ∗n (w)Θn (w) Ln+1 dn (w) −1 −∗ = Ln+1 (w) = dn (w) Ln+1 dn (w) and hence

5 −1 5 Ln+1 I − |w|2 Θ∗n (w)Θn (w) Ln+1 = Ln+1 . Therefore, taking into account (3.5) and Remark 6.6 we can conclude that −1 5 5 5 −1 5 + Ln+1 = Ln+1 I − |w|2 Θ∗n (w) Rn+1 ˜b[n] Ln+1 Ln+1 n (w) dn (w) −1 5 5 −1 + = I − |w|2 Ln+1 Θ∗n (w) Rn+1 ˜b[n] Ln+1 . n (w) dn (w)

This yields

5 5 −1 + Ln+1 = I − |w|2 Ln+1 Θ∗n (w) Rn+1 ˜b[n] Ln+1 (w) d (w) n n 5 5 2 ∗ = Ln+1 − |w| Ln+1 Θn (w)Θn (w) Ln+1

and consequently

5 5 |w|2 Ln+1Θ∗n (w)Θn (w) Ln+1 = 0. Thus, in view of (3.5), we get 5 5 5 ∗ +5 + wΘn (w) (wΘn (w)) = |w|2 Ln+1 Ln+1 Θ∗n (w)Θn (w) Ln+1 Ln+1 = 0.

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Consequently, (ii) holds. (ii) ⇒ (iv): This follows immediately from (1.13). (iv) ⇒ (vi): By deﬁnition of Bn+1 (w) and Rn+1 (w) we see that (iv) leads to (vi). (vi) ⇒ (ii): This can be proved analogously to the implication ”(v) ⇒ (ii)”. We omit the details. Now we consider a particular situation, where one of the equivalent conditions stated in Theorem 7.1 is satisﬁed. Proposition 7.2. Let n ∈ N and let (Γj )n−1 eodory sequence. Let j=0 be a q × q Carath´ the matrices Ln , Rn , and Mn be deﬁned by (1.4), (1.6), and (6.4). Furthermore, let w ∈ D and let 5 ∗ 5 Γn := 2Mn + 2Ln wΘn−1 (w) 2Rn . (7.3) Then (Γj )nj=0 is q × q Carath´eodory sequence, Θn (w) = 0 holds, and the central q × q Carath´eodory function Ωc,n corresponding to (Γj )nj=0 fulﬁlls Ωc,n(w) = Mn+1 (w).

(7.4)

Moreover, if is a nondegenerate q ×q Carath´eodory sequence, then (Γj )nj=0 is a nondegenerate q × q Carath´eodory sequence as well. ∗ Proof. Proposition 3.4 yields that matrix wΘn−1 (w) is strictly contractive. Consequently, from [DFK, Theorem 3.4.1] we get that (Γj )nj=0 is a q × q Carath´eodory sequence and, moreover, that (Γj )nj=0 is a nondegenerate q × q Carath´eodory sequence if (Γj )n−1 eodory sequence. Because of j=0 is a nondegenerate q × q Carath´ (3.4) and (7.3) we have ∗ 5 + 1 5 + Γ − Mn Rn . wΘn−1 (w) = Ln 2 n (Γj )n−1 j=0

Thus application of Theorem 6.8 provides us Θn (w) = 0. Hence using Theorem 7.1 we get (7.4). In the nondegenerate case, the statement of Proposition 7.2 (respectively, Theorem 7.1) can be extended as follows. Proposition 7.3. Let n ∈ N, let (Γj )nj=0 be a nondegenerate q × q Carath´eodory sequence, let Ωc,n be the central q × q Carath´eodory function corresponding to (Γj )nj=0 , and let w ∈ D \ {0}. Then (7.4) is equivalent to 5 ∗ 5 Γn = 2Mn + 2Ln wΘn−1 (w) 2Rn . (7.5) Proof. By virtue of Proposition 7.2 we see that (7.5) implies (7.4). We suppose now (7.4). Because of Theorem 7.1 and w = 0 it follows Θn (w) = 0. Since the q × q Carath´eodory sequence (Γj )nj=0 is nondegenerate, from [FKL1, Lemma 5.1] we know that the matrices Ln+1 , Ln , Rn+1 , and Rn are positive Hermitian. Using this in combination with Θn (w) = 0 and Theorem 6.8 we get then the identity 0 = wΘn−1 (w) − Kn∗ √ √ and, in view of (6.5) and Γn ∈ K 2Mn ; 2Ln , 2Rn , consequently (7.5).

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Let n ∈ N, let (Γj )nj=0 be a q × q Carath´eodory sequence with nonsingular matrix Γ0 , and let Ωc,n be the central q × q Carath´eodory function corresponding to (Γj )nj=0 . Then det Ωc,n does not vanish in D (see [FK1, Remark 30 in Part I]) and the central q × q Carath´eodory function Ωc,n corresponding to the reciprocal q × q Carath´eodory sequence (Γj )nj=0 corresponding to (Γj )nj=0 coincides with the matrix-valued function (Ωc,n )−1 (see [FK3, Theorem 3.3]). Taking this into account we get the following results which are analogues of Theorem 7.1, Proposition 7.2, and Proposition 7.3 with respect to the Weyl matrix balls stated in Theorem 5.2. Here we use again the notations of Theorem 5.2 and Remark 5.3. Theorem 7.4. Let n ∈ N0 , let (Γj )nj=0 be a q × q Carath´eodory sequence with nonsingular matrix Γ0 , let Ωc,n be the central q × q Carath´eodory function corresponding to (Γj )nj=0 , and let w ∈ D. Furthermore, let the matrices Ln+1 and Rn+1 be deﬁned by (1.4) and (1.6). Then the following statements are equivalent: −1 (i) Ωc,n (w) = Mn+1 (w). (ii) wΘn (w) = 0. −1 5 Ln+1 . (iii) An+1 (w) = cn (w) 5 −1 . (iv) Bn+1 (w) = Rn+1 an (w) −1 −∗ Ln+1 cn (w) . (v) Ln+1 (w) = cn (w) −1 −∗ Rn+1 an (w) . (vi) Rn+1 (w) = an (w) Proof. (i) ⇔ (ii): If w = 0, the equivalence of (i) and (ii) is an easy consequence of (7.2) and (5.15). Now suppose w = 0. Moreover, let (Γj )nj=0 be the reciprocal q × q Carath´eodory sequence corresponding to (Γj )nj=0 , let an , bn , cn , and dn be the matrix polynomials given by (5.3), (5.4), (5.5), and (5.6), and let Θn be the rational matrix-valued function deﬁned by (5.8). In view of Proposition 3.4 (with ∗ respect to the q × q Carath´eodory sequence (Γj )nj=0 ) we see that −wΘn (w) is contractive. Because of (7.2), Remark 3.7, and (5.7), on the one hand we have −1 −1 −1 = bn (w) an (w) = an (w) bn (w) Ωc,n (w) −1 [n] [n] 6 6 = −w(cn ) (w)F1• (w) + an (w) w(dn ) (w)F1• (w) + bn (w) 7 + 7 with F1• (w) := Ln+1 0 Rn+1 , where on the other hand (5.24) and (5.19) imply −1 6 [n] 6 [n] Mn+1 (w) = −w(cn ) (w)F2• (w) + an (w) w(dn ) (w)F2• (w) + bn (w) 7 + ∗ 7 with F2• (w) := Ln+1 −wΘn (w) Rn+1 . Therefore, in view of w = 0 and part (a) of Proposition 2.2, we get that (i) is necessary and suﬃcient for ∗ Ln+1 (Ln+1 )+ −wΘn (w) Rn+1 (Rn+1 )+ = Ln+1 (Ln+1 )+ 0Rn+1 (Rn+1 )+ , i.e., (i) is equivalent to

Rn+1 (Rn+1 )+ wΘn (w) Ln+1 (Ln+1 )+ = 0.

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Thus, because of (5.8), statement (i) is tantamount to wΘn (w) = 0.

(7.6)

wΘn (w)

Since = 0 leads immediately to (7.6) in view of (5.12), we obtain that (ii) gives rise to (i). Conversely, if (i) is satisﬁed, then (7.6) holds and hence 7 wΘn (w) Ln+1 = 0. By virtue of (5.11) and (5.12) it follows then 7 5 0 = wV Θn (w)U Ln+1 = wV Θn (w) Ln+1 Γ−∗ 0 , 5 i.e., we get wΘn (w) Ln+1 = 0, which results in wΘn (w) = 0 due to (3.9). Consequently, (i) also implies (ii) in that case. The remaining part of the assertion can be analogously proved as Theorem 7.1 by a straightforward calculation. We omit the details. Now we consider a particular situation, where one of the equivalent conditions stated in Theorem 7.4 is satisﬁed. Proposition 7.5. Let n ∈ N and let (Γj )n−1 eodory sequence with j=0 be a q × q Carath´ nonsingular matrix Γ0 . Furthermore, let w ∈ D and let 5 ∗ 5 Γn := 2Mn + 2Ln −wΘn−1 (w) 2Rn . Then (Γj )nj=0 is a q × q Carath´eodory sequence, Θn (w) = 0 holds, and the central q × q Carath´eodory function Ωc,n corresponding to (Γj )nj=0 fulﬁlls −1 = Mn+1 (w). Ωc,n(w) eodory sequence, then (Γj )nj=0 Moreover, if (Γj )n−1 j=0 is a nondegenerate q ×q Carath´ is a nondegenerate q × q Carath´eodory sequence as well. Proof. Applying Proposition 3.8, [DFK, Theorem 3.4.1], Theorem 6.11, and Theorem 7.4 one can prove Proposition 7.5 analogously to Proposition 7.2. In the nondegenerate case, the statement of Proposition 7.5 (respectively, Theorem 7.4) can be extended as follows. Proposition 7.6. Let n ∈ N and let (Γj )nj=0 be a q × q nondegenerate Carath´eodory sequence, let Ωc,n be the central q × q Carath´eodory function corresponding to −1 (Γj )nj=0 , and let w ∈ D \ {0}. Then Ωc,n (w) = Mn+1 (w) is equivalent to 5 ∗ 5 2Rn . (7.7) Γn = 2Mn + 2Ln −wΘn−1 (w) Proof. Using Proposition 7.5, Theorem 7.4, [FKL1, Lemma 5.1], and Theorem 6.11 one can prove Proposition 7.6 analogously to Proposition 7.3. The following example illustrates that, in the case that the q×q Carath´eodory sequence (Γj )nj=0 is degenerate, equation (7.5) (respectively, (7.7)) is not necessary for the fact that Θn (w) = 0 (respectively, Θn (w) = 0) holds for some w ∈ D.

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Example 7.7. Let U be a unitary q ×q matrix, let Γ0 := Iq , and let Γ1 := 2U . Then (Γj )1j=0 is a q × q Carath´eodory sequence with L1 = R1 = Iq , L2 = R2 = 0q×q , and det Γ0 = 0. In view of (3.4) and (3.10) we have Θ1 (w) = 0 and Θ1 (w) = 0 for each w ∈ D. On the other hand, 5 ∗ 5 2R1 Γ1 = 2U = 2wIq = 2M1 + 2L1 wΘ0 (w) and Γ1 = 2U = −2wIq = 2M1 +

5

∗ 5 2L1 −wΘ0 (w) 2R1 .

We ﬁnish this paper with some comments on central q × q Carath´eodory functions, which are conclusions of certain considerations carried out above. Firstly, we will point out that Proposition 2.2 leads to a characterization of the fact that given ﬁnite q × q Carath´eodory sequences coincide in terms of equality of some value of the corresponding central q × q Carath´eodory functions. The following result paves the way for a proof of this characterization. Lemma 7.8. Let n ∈ N0 and let (Γj )nj=0 be a q × q Carath´eodory sequence. If n ≥ 1, then let Vn ∈ Y n and let Wn ∈ Z n . Let the matrix polynomials an , bn , cn , and dn be deﬁned by (1.7), (1.8), (1.9), and (1.10). Let the matrices Ln+1 and Rn+1 be given by (1.4) and (1.6). Furthermore, for j ∈ {1, 2}, let Kj be a contractive 5 5 + q × q matrix and let Fj := Ln+1 Kj Rn+1 . Then the complex q × q matrices [n] [n] z d˜n (z)F1 + bn (z) and z d˜n (z)F2 + bn (z) are both nonsingular for each z ∈ D and, moreover, the following statements are equivalent: + + + (i) Ln+1 L+ n+1 K1 Rn+1 Rn+1 = Ln+1 Ln+1 K2 Rn+1 Rn+1 . (ii) For each z ∈ D, −1 z d˜n[n] (z)F1 + bn (z) −z˜ c[n] n (z)F1 + an (z) −1 = −z˜ c[n] z d˜n[n] (z)F2 + bn (z) . (7.8) n (z)F2 + an (z) (iii) There is some z ∈ D \ {0} such that (7.8) is satisﬁed. Proof. Apply part (a) of Proposition 2.2.

n+1 Proposition 7.9. Let n ∈ N0 , let (Γj )n+1 eodory sej=0 and (Γj )j=0 be q × q Carath´ k for each k ∈ N0,n , and let Ωc,n+1 and Ω c,n+1 be the quences such that Γk = Γ n+1 central q × q Carath´eodory function corresponding to (Γj )n+1 j=0 and (Γj )j=0 , respec n+1 if and only if there is some z ∈ D \ {0} such that tively. Then Γn+1 = Γ c,n+1 (z). Ωc,n+1 (z) = Ω (7.9) n+1 is fulﬁlled, then the q × q Carath´eodory seProof. If the equality Γn+1 = Γ n+1 n+1 quences (Γj )j=0 and (Γj )j=0 coincide. Thus, by deﬁnition we have fc,n+1 = f c,n+1 . In particular, there is some z ∈ D \ {0} such that the identity (7.9) holds. Conversely, we assume now that (7.9) is satisﬁed for some z ∈ D \ {0}. Because of n n+1 eodory the choice of (Γj )n+1 j=0 and (Γj )j=0 , it follows that (Γj )j=0 is a q × q Carath´

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k for each k ∈ N0,n . Hence, from [DFK, Theorem 3.4.1] we sequence, where Γk = Γ obtain that there are some contractive q × q matrices K1 and K2 fulﬁlling 5 5 Γn+1 = 2Mn+1 + 2Ln+1 K1 2Rn+1 (7.10) and

5 5 n+1 = 2Mn+1 + 2Ln+1K2 2Rn+1 , (7.11) Γ where Ln+1 , Rn+1 , and Mn+1 are deﬁned as in (1.4), (1.6), and (6.4). Therefore, if the matrix polynomials an , bn , cn , and dn are given by (1.7), (1.8), (1.9), and (1.10) with some Vn ∈ Y n and Wn ∈ Z n if n ≥ 1, then (7.9) and [FKL1, Corollary 2.7] 5 5 + yield (7.8) for some z ∈ D \ {0}, where Fj := Ln+1 Kj Rn+1 for j ∈ {1, 2}. By virtue of Lemma 7.8 we get + + + Ln+1 L+ n+1 K1 Rn+1 Rn+1 = Ln+1 Ln+1 K2 Rn+1 Rn+1

n+1 in view of (7.10) and (7.11). which implies ﬁnally Γn+1 = Γ

We note that, since the central q × q Carath´eodory function corresponding ˜ j )0 is the constant function (deﬁned on D) to a q × q Carath´eodory sequence (Γ j=0 ˜ with value Γ0 (see, e.g., [FK3, Remark 1.1]), the assertion of Proposition 7.9 is ˜ j )0 . obvious in the case of given q × q Carath´eodory sequences (Γj )0j=0 and (Γ j=0 Let Ω ∈ Cq (D). Then the matricial version of the F. Riesz-Herglotz Representation Theorem (see, e.g., [DFK, Theorem 2.2.2]) shows that there is a unique nonnegative Hermitian q × q Borel measure F deﬁned on the σ-algebra BT of all Borel subsets of the unit circle T such that 4 z+w Ω(w) = F (dz) + i Im Ω(0) (7.12) T z−w for each w ∈ D. This nonnegative Hermitian measure F is called the Riesz-Herglotz measure of Ω. Let λ be denote the linear Lebesgue measure deﬁned on BT . If Γ0 is a complex q × q matrix with nonnegative Hermitian real part Re Γ0 , then from [FK3, Remark 1.1] and the matricial versions of the F. Riesz-Herglotz Representation Theorem and the Herglotz-Bochner Theorem (see, e.g., [DFK, Theorem 2.2.1]) we know that the Riesz-Herglotz measure Fc,0 of the central q × q Carath´eodory function Ωc,0 corresponding to (Γj )0j=0 is absolutely continuous with respect to 1 λ and that f0 : T → Cq×q deﬁned by f0 (z) := Re Γ0 is a version of the corres2π ponding Radon-Nikodym derivative. In the case that a positive integer n and a nondegenerate q × q Carath´eodory sequence (Γj )nj=0 are given the Riesz-Herglotz measure Fc,n of the central q × q Carath´eodory function Ωc,n corresponding to 1 λ and the corresponding (Γj )nj=0 is also absolutely continuous with respect to 2π Radon-Nikodym derivative can be constructed explicitly from the given sequence (Γj )nj=0 as well (see, e.g., [FK1, Theorem 16 and Remark 18 in Part III]). Let n ∈ N, let (Γj )nj=0 be a nondegenerate q × q Carath´eodory sequence, + + let Vn := Tn−1 Yn , let Wn := Zn Tn−1 , and let the q × q matrix polynomials bn and dn be deﬁned by (1.8) and (1.10), respectively. From [FK3, Proposition 2.2

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and Theorem 2.3] we know that the functions det bn and det dn vanish nowhere in D ∪ T. Thus the following proposition is a generalization of the corresponding result for the nondegenerate situation. Proposition 7.10. Let n ∈ N and let (Γj )nj=0 be a q × q Carath´eodory sequence. Let Vn ∈ Y n , let Wn ∈ Z n , and let the matrix polynomials bn and dn be deﬁned by (1.8) and (1.10). Furthermore, let the matrices Ln+1 and Rn+1 be given by (1.6). (a) If det bn (z) = 0 for each z ∈ T, then the Riesz-Herglotz measure Fc,n of the central q×q Carath´eodory function Ωc,n corresponding to (Γj )nj=0 is absolutely 1 λ and f : T → Cq×q given by continuous with respect to 2π −∗ −1 Rn+1 bn (z) f (z) := bn (z) is the corresponding Radon-Nikodym derivative. (b) If det dn (z) = 0 for each z ∈ T, then Fc,n is absolutely continuous with respect 1 to 2π λ and g : T → Cq×q given by −1 −∗ Ln+1 dn (z) g(z) := dn (z) is the corresponding Radon-Nikodym derivative. Proof. (a) Suppose that det bn does not vanish in T. Since Vn belongs to Y n and since det bn is a polynomial, we get that there is a real number ρ > 1 such that −1 det bn has no zero in K(0; ρ). Hence Ω c,n := an bn is a rational matrix function which is holomorphic in K(0; ρ). Using the matricial version of the H.A. Schwarz Formula, for each w ∈ D, we get 4 1 z+w Re Ωc,n (z) λ(dz) + i Im Ω Ω (w) = c,n c,n (0), 2π T z − w i.e., in view of (7.2) and Ωc,n ∈ Cq [D, (Γj )nj=0 ],

4 −1 −∗ ∗ 1 z + w 1 + bn (z) an (z) λ(dz) + i Im Γ0 . Ωc,n (w) = an (z) bn (z) 2π T z − w 2 Furthermore, from [FKL1, part (a) of Proposition 2.4] we obtain that Re a∗n (z)bn (z) = Rn+1 holds for each z ∈ T. Thus we can conclude that 4 −∗ −1 1 z + w bn (z) Ωc,n (w) = Rn+1 bn (z) λ(dz) + i Im Γ0 2π T z − w

(7.13)

is satisﬁed for each w ∈ D. Taking into account that the complex q × q matrix Rn+1 is nonnegative Hermitian (see, e.g., [DFK, Lemma 1.1.9]) a comparison of (7.12) and (7.13) completes the proof of part (a). (b) This can be proved analogously to (a). In view of Proposition 7.10, in the following result we use the notation Y)n for the set of all Vn ∈ Yn such that det bn vanishes nowhere in D ∪ T and the notation

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Z)n for the set of all Wn ∈ Zn such that det dn vanishes nowhere in D ∪ T, where bn and dn are the matrix polynomials deﬁned by (1.8) and (1.10). Corollary 7.11. Let n ∈ N and let (Γj )nj=0 be a q × q Carath´eodory sequence. Let Vn ∈ Yn , let Wn ∈ Zn , and let the q ×q matrix polynomials bn and dn be deﬁned by (1.8) and (1.10). Furthermore, let the matrices Ln+1 and Rn+1 be given by (1.6). If at least one of the sets Y)n and Z)n is nonempty, then the Riesz-Herglotz measure Fc,n of the central q × q Carath´eodory function Ωc,n corresponding to (Γj )nj=0 is 1 λ and the corresponding Radon-Nikodym absolutely continuous with respect to 2π q×q derivative f : T → C satisﬁes λ-a.e. on T the identities −1 f = b−∗ n Rn+1 bn

and

−∗ f = d−1 n Ln+1 dn .

Proof. An application of Proposition 7.10 in combination with Remark 3.1 and part (a) of Lemma 3.2 yields the assertion. Proposition 7.12. Let n ∈ N, let (Γj )nj=0 be a q × q Carath´eodory sequence with nonsingular matrix Γ0 , and let Ωc,n the central q × q Carath´eodory function corresponding to (Γj )nj=0 . Let Vn ∈ Y n , let Wn ∈ Z n , and let the matrix polynomials an and cn be deﬁned by (1.7) and (1.9). Furthermore, let the matrices Ln+1 and Rn+1 be given by (1.6). of Ω−1 (a) If det an (z) = 0 for each z ∈ T, then the Riesz-Herglotz measure Fc,n c,n 1 q×q is absolutely continuous with respect to 2π λ and f : T → C given by −∗ −1 Rn+1 an (z) f (z) := an (z) is the corresponding Radon-Nikodym derivative. (b) If det cn (z) = 0 for each z ∈ T, then Fc,n is absolutely continuous with respect 1 q×q given by to 2π λ and g : T → C −1 −∗ g (z) := cn (z) Ln+1 cn (z) is the corresponding Radon-Nikodym derivative. Proof. Based on Remark 3.7, the matricial version of the H.A. Schwarz Formula, (7.2), and [FKL1, part (a) of Proposition 2.4] one can prove Proposition 7.12 analogously to Proposition 7.10. We omit the details. In view of Proposition 7.12 and Remark 3.7 we use in the following result the notation Yˇn for the set of all Vn ∈ Yn such that det an vanishes nowhere in D ∪ T and the notation Zˇn for the set of all Wn ∈ Zn such that det cn vanishes nowhere in D ∪ T, where an and cn are the matrix polynomials deﬁned by (1.7) and (1.9). Corollary 7.13. Let n ∈ N, let (Γj )nj=0 be a q × q Carath´eodory sequence with nonsingular matrix Γ0 , and let the matrices Ln+1 and Rn+1 be given by (1.6). Furthermore, let Vn ∈ Yn , let Wn ∈ Zn , and let the q×q matrix polynomials an and cn be deﬁned by (1.7) and (1.9). If at least one of the sets Yˇn and Zˇn is nonempty, then the Riesz-Herglotz measure Fc,n of Ω−1 c,n is absolutely continuous with respect

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1 λ and the corresponding Radon-Nikodym derivative f : T → Cq×q satisﬁes to 2π λ-a.e. on T the identities −1 f = a−∗ n Rn+1 an

and

−∗ f = c−1 n Ln+1 cn .

Proof. An application of Proposition 7.12 in combination with Remark 3.1 and part (b) of Lemma 3.2 yields the assertion.

References [Akh]

Akhiezer, N.I.: The Classical Moment Problem (Russian), Fizmatgiz, Moskva 1961; English translation: Oliver and Boyd, Edinburgh-London 1965. [AK] Arov, D.Z.; Kre˘ın, M.G.: Problems of the search of the minimum of entropy in indeterminate extension problems (Russian), Funkcional. Anal. Prilozen. 15 (1981), No. 2, 61–64; English translation: Func. Anal. Appl. 15 (1981), 123–126. [BGR] Ball, J.A.; Gohberg, I.; Rodman, L.: Interpolation of Rational Matrix Functions, Operator Theory: Adv. Appl. 45, Birkh¨ auser, Basel 1990. [BH] Ball, J.A.; Helton, J.W.: Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: parametrization of the set of all solutions, Integral Equations Operator Theory 9 (1986), 155–203. [BD] Bolotnikov, V.; Dym, H.: On degenerate interpolation, entropy and extremal problems for matrix Schur functions, Integral Equations Operator Theory 32 (1998), 367–435. [Br] Bruinsma, P.: Degenerate interpolation problems for Nevanlinna pairs, Indag. Math., N.S. 2 (1991), 179–200. [CG] Chang, C.; Georgiou, T.: Geometric aspects of the Carath´ eodory extension problem, Linear Algebra Appl. 203/204 (1994), 209–251. [CH1] Chen, G.N.; Hu, Y.J.: The truncated Hamburger matrix moment problems in the nondegenerate and degenerate cases, and matrix continued fractions, Linear Algebra Appl. 277 (1998), 199–236. [CH2] Chen, G.N.; Hu, Y.J.: On the multiple Nevanlinna-Pick matrix interpolation in eodory matrix coeﬃcient problem, Linear Algebra the class Cp and the Carath´ Appl. 283 (1998), 179–203. [DGK1] Delsarte, P.; Genin, Y.; Kamp, Y.: Orthogonal polynomial matrices on the unit circle, IEEE Trans. Circuits and Systems CAS-25 (1978), 149–160. [DGK2] Delsarte, P.; Genin, Y.; Kamp, Y.: The Nevanlinna-Pick problem for matrixvalued functions, SIAM J. Appl. Math. 36 (1979), 47–61. [Du] Dubovoj, V.K.: Indeﬁnite metric in the interpolation problem of Schur for analytic matrix functions IV (Russian), Teor. Funktsii, Funkts. Anal. i Prilozen. 42 (1984), 46–57; English translation in: Topics in Interpolation Theory (Eds.: H. Dym, B. Fritzsche, V.E. Katsnelson, B. Kirstein), Operator Theory: Adv. Appl. 95, Birkh¨ auser, Basel 1997, pp. 93–104. [DFK] Dubovoj, V.K.; Fritzsche, B.; Kirstein, B.: Matricial Version of the Classical Schur Problem, Teubner-Texte zur Mathematik 129, B.G. Teubner, StuttgartLeipzig 1992.

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Dym, H.: J Contractive Matrix Functions, Reproducing Kernel Spaces and Interpolation, CBMS Regional Conference Series in Mathematics 71, Amer. Math. Soc., Providence, R.I. 1989. [Ev] Everitt, W.N.: A personal history of the m-coeﬃcient, J. Comp. Appl. Math. 171 (2004), 185–197. [FF] Foia¸s, C.; Frazho, A.E.: The Commutant Lifting Approach to Interpolation Problems, Operator Theory: Adv. Appl. 44, Birkh¨ auser, Basel 1990. [FFGK] Foia¸s, C.; Frazho, A.E.; Gohberg, I.; Kaashoek, M.A.: Metric Constrained Interpolation, Commutant Lifting and Systems, Operator Theory: Adv. Appl. 100, Birkh¨ auser, Basel 1998. [FFK] Fritzsche, B.; Fuchs, S.; Kirstein, B.: A Schur type matrix extension problem, Part V, Math. Nachr. 158 (1992), 133–159. [FK1] Fritzsche, B.; Kirstein, B.: An extension problem for non-negative hermitian block Toeplitz matrices, Math. Nachr., Part I: 130 (1987), 121–135; Part II: 131 (1987), 287–297; Part III: 135 (1988), 319–341; Part IV: 143 (1989), 329–354; Part V: 144 (1989), 283–308. [FK2] Fritzsche, B.; Kirstein, B.: On the Weyl matrix balls associated with nondegenerate matrix-valued Carath´ eodory functions, Zeitschr. f¨ ur Analysis und ihre Anwendungen 12 (1993), 239–261. [FK3] Fritzsche, B.; Kirstein, B.: Representations of central matrix-valued Carath´eodory functions in both nondegenerate and degenerate cases, Integral Equations Operator Theory 50 (2004), 333–361. [FKL1] Fritzsche, B.; Kirstein, B.; Lasarow, A.: The matricial Carath´eodory problem in both nondegenerate and degenerate case, in: Operator Theory: Adv. Appl. (Subseries: Linear Operators and Linear Systems) Vol. 165 (Eds.: D. Alpay, I. Gohberg), Birkh¨ auser, Basel 2006, pp. 251–290. [FKL2] Fritzsche, B.; Kirstein, B.; Lasarow, A.: On a class of extremal solutions of the nondegenerate matricial Carath´eodory problem, Analysis 27 (2007), 109–164. [Ge1] Geronimus, Ja.L.: On polynomials orthogonal on the unit circle, on the trigonometric moment problem and on associated functions of classes of Carath´ eodorySchur (Russian), Mat. USSR-Sb. 15 (1944), 99–130. [Ge2] Geronimus, Ja.L.: On the trigonometric moment problem, Ann. Math. 47 (1946), 742–761. [Ge3] Geronimus, Ja.L.: Polynomials orthogonal on a circle and their applications (Russian), Zapiski Nauˇcno-Issled. Mat. Meh. Har’kov. Mat. Obˇsˇc. 19 (1948), 35–120; English translation in: AMS Translations, Series 1, Volume 3, AMS, Providence, R.I. 1962, pp. 1–78. Gohberg, I.Ts.; Heinig, G.: Inversion of ﬁnite Toeplitz matrices consisting of [GH] elements of a non-commutative algebra (Russian), Rev. Roum. Math. Pures et. Appl. 19 (1974), 623–663. [He] Hellinger, E.: Zur Stieltjesschen Kettenbruchtheorie, Math. Ann. 86 (1922), 18–29. [Ko] Kovalishina, I.V.: Analytic theory of a class of interpolation problems (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 47 (1983), 455–497; English translation: Math. USSR Izvestija 22 (1984), 419–463.

332 [KP]

[Mi]

[Ne] [Or]

[Sa] [Sm]

[We]

B. Fritzsche, B. Kirstein and A. Lasarow Kovalishina, I.V.; Potapov, V.P.: The radii of a Weyl disk in the matricial Nevanlinna-Pick problem (Russian), in: Operator Theory in Function Spaces and Their Application (Ed.: V.A. Marˇcenko), Naukova Dumka, Kiev 1981, pp. 25–49; English translation in: Seven Papers Translated from the Russian, AMS Translations, Series 2, Volume 138, AMS, Providence, R.I. 1988, pp. 37–54. Mikhailova, I.V.: Weyl matrix circles as a tool for uniqueness in the theory of multiplicative representations of J-inner matrix functions (Russian), in: Analysis in Inﬁnite-dimensional Spaces and Operator Theory (Ed.: V.A. Marˇcenko), Naukova Dumka, Kiev 1983, pp. 101–117; English translation in: Topics in Interpolation Theory (Eds.: H. Dym, B. Fritzsche, V.E. Katsnelson, B. Kirstein), Operator Theory: Adv. Appl. 95, Birkh¨ auser, Basel 1997, pp. 397–417. Nevanlinna, R.: Asymptotische Entwicklungen beschr¨ ankter Funktionen und das Stieltjessche Momentenproblem, Ann. Acad. Sci. Fenn. A18 (1922), 5, 1–53. Orlov, S.A.: Nested matrix balls, analytically depending on a parameter, and theorems on invariance of ranks of radii of limit matrix balls (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 40 (1976), 593–644; English translation: Math. USSR Izvestija 10 (1976), 565–613. Sakhnovich, L.A.: Interpolation Theory and its Applications, Mathematics and its Applications 428, Kluwer, Dordrecht 1997. ˇ Smuljan, Ju.L.: Operator balls (Russian), Teor. Funkcii, Funktional. Anal. i Prilozen. 6 (1968), 68–81; reprinted in: Integral Equations and Operator Theory 13 (1990), 864–882. ¨ Weyl, H.: Uber gew¨ ohnliche Diﬀerentialgleichungen mit Singularit¨ aten und die zugeh¨ origen Entwicklungen willk¨ urlicher Funktionen, Math. Ann. 68 (1910), 220–269.

Bernd Fritzsche and Bernd Kirstein Fakult¨ at f¨ ur Mathematik und Informatik Universit¨ at Leipzig Postfach: 10 09 20 D-04009 Leipzig, Germany e-mail: [email protected] [email protected] Andreas Lasarow Departement Computerwetenschappen Katholieke Universiteit Leuven Celestijnenlaan 200A – postbus: 02402 B-3001 Leuven, Belgium e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 199, 333–346 c 2010 Birkh¨ auser Verlag Basel/Switzerland

On Extremal Problems of Interpolation Theory with Unique Solution Bernd Fritzsche, Bernd Kirstein and Lev A. Sakhnovich Dedicated to the memory of Georg Heinig

Abstract. The main goal of this paper is to investigate the matrix extremal interpolation problem formulated in Chapter 7 of the monograph [7]. We give natural conditions under which the problem has one and only one solution. The basic idea of the proof is to use the matrix Riccati equation deduced in [7, Chapter 7]. Mathematics Subject Classiﬁcation (2000). Primary: 47A57. Keywords. Matricial interpolation problems, minimal solutions, nonlinear matrix equation, regular interpolation problems.

0. Introduction In this paper we consider a particular matrix extremal interpolation problem. More precisely, we try to ﬁnd amongst the solutions w(z) of the corresponding interpolation problem the solution which satisﬁes the additional extremal condition w∗ (z)w(z) ≤ ρ2min ,

|z| < 1,

(0.1)

where ρmin is a positive Hermitian m × m matrix. What concerns the statement of the problem we follow the book [7, Chapter 7] where some Riccati type equation for the matrix ρmin was deduced. It was proved by Ran and Reurings (see [6]) that for the case of the Schur extremal interpolation problem this equation has one and only one solution ρmin . The main result of this paper is to show that the above-mentioned result of Ran and Reurings can be extended to a broad class of interpolation problems. In particular, it is true for the Nevanlinna-Pick problem. In this paper we will develop a method for the computation of ρmin . We illustrate our general result with some concrete examples (Schur problem, Nevanlinna-Pick problem, Jordan blocks). Remark 0.1. A well-posed physical problem should have one and only one solution. The extremal interpolation problem under consideration satisﬁes this requirement.

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Remark 0.2. It is essential both from the applied and theoretical points of view that the solution of the extremal problem turns out to be a rational matrix function (see [7, Chapter 7], [4]). Remark 0.3. It is possible that in the classical case w∗ (z)w(z) ≤ I the interpolation problem has not any solution whereas problem (0.1) has a solution. Remark 0.4. The scalar case (m = 1) of the extremal interpolation problem was studied by N.I. Akhiezer [1]. It found its application in control theory (see Kimura [5]). The transition to the matrix case allows to enlarge considerably the class of extremal problems which have eﬀective solutions.

1. Extremal interpolation problems Let the matrices A, Sk and Ψk , k = 1, 2, have the sizes mN × mN and mN × m, respectively, where Sk is nonnegative Hermitian. We suppose that these matrices are connected by the relations Sk − ASk A∗ = Ψk Ψ∗k ,

k = 1, 2.

(1.1)

Setting S := S2 − S1 we deduce from (1.1) the equality S − ASA∗ = Ψ2 Ψ∗2 − Ψ1 Ψ∗1 .

(1.2)

We introduce the block-diagonal matrix R := diag (ρ, . . . , ρ) N

where ρ is a positive Hermitian matrix of size m × m. In addition we shall assume the equality AR = RA. (1.3) This is justiﬁed, since it was shown in [4] that condition (1.3) is true in a number of concrete examples. From equations (1.1) and (1.3) it follows that Sρ − ASρ A∗ = Ψ2 Ψ∗2 − Ψ1,ρ Ψ∗1,ρ .

(1.4)

Sρ := S2 − R−1 S1 R−1

(1.5)

where −1

Ψ1,ρ := R Ψ1 . (1.6) Thus we have constructed a set of operator identities (1.5), where the positive Hermitian matrix ρ plays the role of a parameter. A set of interpolation problems, see

On Extremal Problems of Interpolation Theory

335

[7, Chapter 6] corresponds to this set of operator identities. A necessary condition for the solvability of these problems is the inequality RS2 R − S1 ≥ 0.

(1.7)

Now we turn to extremal interpolation. Deﬁnition 1.1. We shall call the matrix ρ = ρmin > 0 a minimal solution of inequality (1.7) if the following two requirements are fulﬁlled: 1. The inequality (1.8) Rmin S2 Rmin − S1 ≥ 0 holds where Rmin = diag (ρmin , . . . , ρmin ) N

is valid. 2. If ρ > 0 satisﬁes inequality (1.7), then rank (Rmin S2 Rmin − S1 ) ≤ rank (RS2 R − S1 ).

(1.9)

(In other words, Rmin minimizes the rank of RS2 R − S1 ≥ 0.) Remark 1.2. The existence of ρmin follows directly from Deﬁnition 1.1 We shall write the nonnegative Hermitian matrices S1 , S2 and R in the following block forms (k) (k) S11 S12 k = 1, 2 (1.10) Sk = (k) (k) , S21 S22

R1 0 R= (1.11) , R1 = diag (ρ, . . . , ρ) 0 ρ N −1

(k)

(k)

where S22 are blocks of size m × m, S11 has the size (N − 1)m × (N − 1)m and (k) S12 has the size (N − 1)m × m. The following result is proved in [7, Proposition 7.1.1]. Proposition 1.3. Suppose that for all ρ > 0 satisfying inequality (1.7) the upper diagonal block is positive Hermitian, i.e., that (2)

(1)

R1 S11 R1 − S11 > 0 holds. If ρ = q > 0 satisﬁes inequality (1.7) and the relation −1 (2) (1) (2) (1) C1 qS22 q = S22 + C1∗ Q1 S11 Q1 − S11 where Q1 := diag (q, q, . . . , q),

(2)

(1)

C1 := Q1 S12 q − S12 ,

N −1

then ρmin = q.

(1.12) (1.13)

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2. On a nonlinear matrix equation In this section we consider the equation −1 (2) (1) (2) (1) ∗ S12 qS22 q = S22 + S12 Q1 S11 Q1 − S11

(2.1)

where (2)

(1)

S12 := Q1 S12 q − S12 .

(2.2)

We make the following two assumptions. Condition 1. The matrix S2 has the block structure S2 = (Cjk )nj,k=1 where all m × m blocks Cjk have the shape Cjk = αjk Im with some complex number αjk . Condition 2. The matrix S2 is positive Hermitian (S2 > 0). In view of Condition 1 we obtain S12

(2)

(1)

= Q21 S12 − S12 −1 (2) (2) (2) (1) = Q21 S11 S11 S12 − S12 ( −1 −1 ' (2) (1) (2) (2) (1) (2) (2) (1) S11 = Q21 S11 − S11 S12 + S11 S11 S12 − S21

and (2)

(2)

Q1 S11 Q1 = Q21 S11 . We introduce the following notations (2)

(1)

A := Q21 S11 − S11 , and (1)

C := S11

−1 (2) (2) B := S11 S12 ,

−1 (2) (2) (1) S12 − S12 . S11

(2.3)

(2.4)

Then obviously A∗ = A and the equation (2.1) can be written in the form αnn q 2 = S22 + (B ∗ A + C ∗ )A−1 (AB + C) (1)

or αnn q 2 = S22 + B ∗ AB + B ∗ C + C ∗ B + C ∗ A−1 C. (1)

(2.5)

Using (2.3) and (2.5) we infer

where

q 2 T = U + C ∗ A−1 C

(2.6)

∗ −1 (2) (2) (2) S11 S12 T := αnn Im − S12

(2.7)

On Extremal Problems of Interpolation Theory

337

and U

:= =

B ∗ C + C ∗ B − B ∗ S11 B + S22 ∗ −1 −1 ∗ −1 (2) (2) (1) (2) (2) (2) (2) (1) S12 S11 S11 S12 − S12 S12 S11 S11 ∗ −1 (1) (2) (2) (1) − S12 S12 + S22 (2.8) S11 (1)

(1)

In view of Condition 2 the relation T >0

(2.9)

is true. According to Condition 1 and (2.7) the matrix T has scalar type, i.e., T = βIm .

(2.10)

Because of (2.9) and (2.10) it follows then β > 0. Hence the equation (2.6) takes the form q2 =

1 (U + C ∗ A−1 C). β

(2.11)

If we compare the equations (2.1) and (2.11) we see that S12 depends on q, but C does not depend on q. Taking into account this fact we can apply Theorem 3.3 of the paper [6] to equation (2.11). Moreover, we observe that the matrix A can be represented in the form (1) A = DQ21 D − S11 where D :=

7 (2) S11 > 0.

Now we rewrite the equation (2.11) in the form

−1 1 (1) U + C1∗ Q21 − D−1 S11 D−1 q2 = C1 β where C1 := D−1 C. We introduce the notation ˜ := diag (U, U, . . . , U ) . U n−1

Deﬁnition 2.1. We call an interpolation problem regular if the condition 1˜ (1) U > D−1 S11 D−1 β is satisﬁed.

(2.12)

338

B. Fritzsche, B. Kirstein and L.A. Sakhnovich Using Theorem 3.3 in [6] now we obtain our main result.

Theorem 2.2. Let the conditions 1 and 2 be fulﬁlled and let the interpolation prob(1) lem be regular. If S11 ≥ 0 then equation (2.1) has a unique solution q such that (2) (1) q > 0 and Q1 S11 Q1 > S11 . Corollary 2.3. Under the assumptions of Theorem 2.2 the relation ρ2min = q 2 holds. Remark 2.4. Under the assumption that ρmin is known an explicit representation of the solution of the corresponding extremal interpolation problem is given in monograph [7, Chapter 7]. In the special case of the extremal interpolation problems named after Schur and Nevanlinna-Pick, respectively, this solution is written in a simpler form in the paper [4]. We have shown that there is one and only one positive Hermitian solution of equation (2.1) which satisﬁes condition (1.7). In the case N = 2 Ferrante/Levy [3] proved without taking into account condition (1.7) that equation (2.1) has one and only one positive Hermitian solution.

3. Methods of computation 1. We apply the method of successive approximation to the study of equation (2.1). We set 1 q02 := U β and for p ∈ {0, 1, 2, . . .} then 1 −1 −1 2 := D )C1 , qp+1 U + C1∗ (Q2p − D−1 S11 β where Qp := diag (qp , qp , . . . , qp ) . N −1

The following assertion is proved in [7, Lemma 7.1.1]. Lemma 3.1. Let the conditions of Theorem 2.2 be fulﬁlled. Then: a) The sequence q02 , q22 , q42 , . . . is monotonously increasing and has the limit q 2 . b) The sequence q12 , q32 , q52 , . . . is monotonously decreasing and has the limit q 2 . The combination of Theorem 2.2 and Lemma 3.1 yields the following result. Corollary 3.2. The relations q 2 = q 2 = ρ2min and 2 2 ≤ ρ2min ≤ q2p+1 , p ∈ {0, 1, 2, . . .}, q2p

hold.

On Extremal Problems of Interpolation Theory

339

2. Now we introduce the matrix Smin := Rmin S2 Rmin − S1 .

(3.1)

R1 := diag (ρmin , . . . , ρmin ) .

(3.2)

Setting N −1

and, using the block partitions (1.10), moreover (2)

(1)

S12 := R1 S12 ρmin − S12

(3.3)

from (3.1), (3.2), and (3.3) we obtain the block decomposition (2) (1) R1 S11 R1 − S11 S12 Smin = (2) (1) . ∗ S12 ρmin S22 ρmin − S22

(3.4)

Taking into account (2.1), (2.2), (3.2), and (3.3) from Theorem 2.2 we get (2)

(1)

R1 S11 R1 S11 > 0

(3.5)

and

∗ ρmin S22 ρmin = S22 + S12 (R1 S11 R1 − S11 )−1 S12 . In view of Smin ≥ 0, (3.4), (3.5) and (3.6) we infer (2)

(1)

(2)

(1)

rankSmin = (N − 1)m.

(3.6) (3.7)

of complex m × m matrices via ⎞

(Yj )nj=1

Deﬁning the sequence ⎛

Y1 ⎜ Y2 ⎟ ⎟ ⎜ (2) (1) ⎜ .. ⎟ := −(R1 S11 R1 − S11 )−1 S12 ⎝ . ⎠ YN −1

(3.8)

and YN := Im we obtain from (3.4) and (3.6) the relation

(3.9)

Smin Y = 0

(3.10)

where Y := col(Yj )N (3.11) j=1 . 3. Let us calculate the number β introduced in (2.10). According to Condi(2) (2) tions 1 and 2 the matrix S22 is nonsingular and the matrix (S22 )−1 has the block structure (2) −1 (S22 )−1 = (Djk )N (3.12) j,k=1 where all m × m blocks Djk have the shape Djk = γjk Im with some complex numbers γjk . Condition 1 implies moreover that (2) −1 S12 = col (αjN Im )N j=1 .

(3.13)

(3.14)

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B. Fritzsche, B. Kirstein and L.A. Sakhnovich

Combining (2.10), (2.7), (3.12), (3.13), and (3.14) we obtain β = αN N − α∗ Γα where −1 α := col(αjN )N j=1

and −1 Γ := (γjk )N j,k=1 .

4. Schur extremal problem Let us consider the following version of the Schur problem. Problem 4.1. Let the complex m × m matrices a0 , a1 , . . . , ap and the positive Hermitian m × m matrix ρ be given. We wish to describe the set of m × m matrix functions w(z) holomorphic in the circle |z| < 1 satisfying w(z) = a0 + a1 z + · · · + ap z p + · · ·

(4.1)

and w∗ (z)w(z) ≤ ρ2 ,

|z| < 1. The relations (4.1) and (4.2) can be written in the form w1 (z) = b0 + b1 z + · · · + bp z p + · · · , w1∗ (z)w1 (z)

≤ Im ,

|z| < 1,

(4.2) (4.3) (4.4)

where w1∗ (z) := w(z) · ρ−1 −1

(4.5) −1

b0 := a0 ρ , . . . , bp := ap ρ . (4.6) Thus, the modiﬁed Problem 4.1 is reduced to the matrix version of the classical Schur problem. Using [7, Proposition 7.2.1] we deduce the following assertion. Proposition 4.2. Let the matrix Cp be ⎛ a0 ⎜0 ⎜ Cp := ⎜ . ⎝ .. 0

deﬁned by a1 a0 .. .

⎞ ... ap . . . ap−1 ⎟ ⎟ .. ⎟ . . ⎠

0

...

(4.7)

a0

Then Problem 4.1 has a solution if and only if the inequality Cp∗ Cp ≤ R2

(4.8)

R := diag (ρ, ρ, . . . , ρ) .

(4.9)

is valid, where p+1

On Extremal Problems of Interpolation Theory

341

Remark 4.3. If the strict inequality Cp∗ Cp < R2

(4.10)

holds then the set of solutions of the modiﬁed Schur problem 4.1 can be written with the aid of a linear fractional transformation (see [7, Chapter 6]). Let us formulate now the extremal Schur problem. Problem 4.4. Let the complex m × m matrices a0 , a1 , . . . , ap be given. We seek an m × m matrix-valued function w(z), holomorphic in the circle |z| < 1, satisfying w(z) = a0 + a1 z + · · · + ap z p + · · ·

(4.11)

and w∗ (z)w(z) ≤ ρ2min ,

|z| < 1.

(4.12)

Here ρmin will be deﬁned by a minimal rank condition. More precisely, we indicate that the Schur extremal problem ﬁts into the general scheme of interpolation problems studied in this paper and seek a minimal rank solution in the sense of Deﬁnition 1.1. We present now the operator reformulation of the Schur problem (see [7, Chapter 7]). It is well known that in this case S2 = I(p+1)m ,

S1 = Cp∗ Cp

(4.13)

where Cp is given by (4.7). Moreover, the matrix A has in the case of the Schur problem the form ⎛

0 ⎜Im ⎜ ⎜ A=⎜0 ⎜ .. ⎝ .

0 0 Im .. .

0

0

... ... ...

0 0 0 .. .

. . . Im

⎞ 0 0⎟ ⎟ 0⎟ ⎟, .. ⎟ .⎠ 0

(4.14)

p+1

whereas the matrices Ψ1 and Ψ2 are deﬁned by formulas Ψ1 = col a∗0 , a∗1 , . . . , a∗p

(4.15)

and Ψ2 = col [Im , 0, . . . , 0] .

(4.16)

Taking into account formulas (4.13)–(4.16) it is immediately checked that Sk − ASk A∗ = Ψk Ψ∗k ,

k ∈ {1, 2}.

(4.17)

Thus, in view of (4.17) the results obtained above can be applied to the Schur extremal problem.

342

B. Fritzsche, B. Kirstein and L.A. Sakhnovich In this case we have T = Im , (2) S12

= 0,

S12 =

β = 1, ∗ S11 = Cp−1 Cp−1 ,

(1) −S12 ,

(1)

U = S22 = a∗0 a0 + a∗1 a1 + · · · + a∗p ap . (1)

The regularity condition from Deﬁnition 2.1 has now the form (1)

diag (U, U, . . . , U ) > S11

(4.18)

p

Proposition 4.5. If the condition (4.18) is fulﬁlled, then a∗p ap > 0. The converse statement is true in the case p = 1. Proposition 4.6. If p = 1 and a∗1 a1 > 0 then U = a∗0 a0 + a∗1 a1 > a∗0 a0 = S11 . (1)

5. Nevanlinna-Pick extremal problem Let us consider the following version of the Nevanlinna-Pick problem. Problem 5.1. Let the complex m × m matrices η1 , η2 , . . . , ηn , the points z1 , . . . , zn satisfying |zk | < 1, k ∈ {1, . . . , n} and the positive Hermitian m × m matrix ρ be given. We wish to describe the set of m × m matrix functions w(z) holomorphic in the circle |z| < 1 such that w(−zk ) = ηk∗ , k ∈ {1, . . . , n},

(5.1)

w∗ (z) · w(z) ≤ ρ2 ,

(5.2)

and |z| < 1.

The relations (5.1) and (5.2) can be written in the form w1 (−zk ) = η˜k , k ∈ {1, . . . , n},

(5.3)

w1∗ (z) · w1 (z) ≤ Im ,

|z| < 1,

(5.4)

η˜k := ηk · ρ−1 , k ∈ {1, . . . , n},

(5.5)

where −1

w1 (z) := w(z) · ρ

.

(5.6)

Using [7, Proposition 7.3.1] we deduce the following assertion. Proposition 5.2. Let the matrices S1 and S2 be deﬁned by

n

n ηk ηl∗ Im , S2 := . S1 := 1 − zk z l k,l=1 1 − zk z l k,l=1

(5.7)

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Then Problem 5.1 has a solution if and only if the inequality S2 − R−1 S1 R−1 ≥ 0

(5.8)

R := diag (ρ, . . . , ρ) .

(5.9)

is valid, where p

Remark 5.3. If the strict inequality S2 − R−1 S1 R−1 > 0 holds, then the set of solutions of the modiﬁed Nevanlinna-Pick Problem 5.1 can be written with the aid of a linear fractional transformation (see [7, Chapter 6]). Let us formulate now the Nevanlinna-Pick extremal problem. Problem 5.4. Let complex m × m matrices η1 , η2 , . . . , ηn and points z1 , z2 , . . . , zn satisfying |zk | < 1, k ∈ {1, . . . , n}, be given. We seek an m × m matrix-valued function w(z) which is holomorphic in the circle |z| < 1, such that and

w(−zk ) = ηk∗ , k ∈ {1, . . . , n},

(5.10)

w∗ (z)w(z) ≤ ρ2min ,

(5.11)

|z| < 1.

Here ρmin will be deﬁned by a minimal rank condition. More precisely, we indicate that the Nevanlinna-Pick extremal problem ﬁts into the general scheme of interpolation problems studied in this paper and seek a minimal rank solution in the sense of Deﬁnition 1.1. We present now the operator reformulation of the Nevanlinna-Pick problem (see [7, Chapter 7]). In this case, the matrices S1 and S2 are given by (5.7) whereas the matrices A, Ψ1 , and Ψ2 are deﬁned by A := diag(z1 Im , z2 Im , . . . , zn Im )

(5.12)

(5.13) Ψ1 := col(η1 , . . . , ηm ) Ψ2 := col(Im , . . . , Im ). (5.14) Taking into account formulas (5.7), (5.12), (5.13) and (5.14) it is immediately checked that (5.15) Sk − ASk A∗ = Ψk Ψ∗k , k ∈ {1, 2}. Thus, in view of (5.15) the results obtained above can be applied to the NevanlinnaPick extremal problem. In this case we have

n−1

n−1 ηk ηl∗ Im (1) (2) S11 = , S11 = , (5.16) 1 − zk z l k,l=1 1 − zk z l k,l=1

(1)

S12

ηn ηn∗ (1) , S22 = 1 − |zn |2

n−1 ηk ηn∗ = , 1 − zk z n k=1

(2)

Im , 1 − |zn |2

n−1 Im = . 1 − zk z n k=1

S22 = (2)

S12

(5.17) (5.18)

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Example. We consider the case n = 2. Then (1)

S11 = (1)

S22 =

η1 η1∗ , 1 − |z1 |2

S11 =

η2 η2∗ , 1 − |z2 |2

S22 =

(2)

(2)

Im 1 − |z1 |2

(5.19)

Im 1 − |z2 |2

(5.20)

η1 η2∗ Im (2) , S12 = . 1 − z1 z 2 1 − z1 z 2 Thus, using formulas (5.16)–(5.21) we obtain (1)

S12 =

β=

(5.21)

1 − |z1 |2 1 − 1 − |z2 |2 (1 − z1 z2 )(1 − z1 z2 ) =

|z1 − z2 |2 . [1 − |z2 |2 ] · |1 − z1 z2 |2

(5.22)

Using (2.8) and (5.22) we get ∗ −1 −1 (2) (2) (1) (2) (2) S11 S11 S11 U = S12 S12 ∗ −1 ∗ −1 (2) (2) (1) (1) (2) (2) (1) S11 S11 − S12 S12 − S12 S12 + S22

1 − |z1 |2 1 1 − |z1 |2 ∗ η2 η2∗ = (η − η )(η − η ) + − 1 2 1 2 |1 − z1 z2 |2 1 − |z2 |2 |1 −z1 z2 | (1 − |z1 |2 )(1 − |z2 |2 ) =β (η1 − η2 )(η1 − η2 )∗ + η2 η2∗ |z1 − z2 |2 Hence, (1 − |z1 |2 )(1 − |z2 |2 ) 1 U= (η1 − η2 )(η1 − η2 )∗ + η2 η2∗ . β |z1 − z2 |2 In view of

7 D=

the regularity condition

1 βU

(2) S11 = 5

Im 1 − |z1 |2

> D−1 S11 D−1 has now the form (1)

(1 − |z1 |2 )(1 − |z2 |2 ) (η1 − η2 )(η1 − η2 )∗ + η2 η2∗ > η1 η1∗ . |z1 − z2 |2 In particular, if η2 η2∗ > η1 η1∗ , then this condition is satisﬁed.

6. Jordan block diagonal structure In this section, we consider the case when the matrix A has the block diagonal form A = diag(A1 , A2 , . . . , AN ),

On Extremal Problems of Interpolation Theory

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where for each k ∈ {1, . . . , N } the complex mnk × mnk matrix Ak has the shape ⎛ ⎞ λk Im 0 ... 0 0 ⎜ Im λk Im . . . 0 0 ⎟ ⎜ ⎟ ⎜ 0 Im ... 0 0 ⎟ Ak = ⎜ (6.1) ⎟ ⎜ .. .. .. .. ⎟ ⎝ . ⎠ . . . 0 0 . . . Im λk Im and where Let n :=

N k=1

|λk | < 1.

(6.2)

nk . Then we choose the complex mn × m matrix Ψ2 via Ψ2 := col (Ψ21 , Ψ22 , . . . , Ψ2,N )

(6.3)

where k ∈ {1, . . . , N }

Ψ2,k := col (Im , 0, . . . , 0),

(6.4)

nk

Now we are looking for a complex mn × mn matrix S2 satisfying the equation S2 − AS2 A∗ = Ψ2 Ψ∗2 .

(6.5)

Proposition 6.1. There is one and only complex mn × mn matrix S2 satisfying (6.5), namely ∞ S2 = Aj Ψ2 Ψ∗2 (Aj )∗ . (6.6) j=0

Moreover, the matrix S2 is positive Hermitian. Proof. In view of (6.1) and (6.2) we see that the spectrum of the matrix A is contained in the open unit disk. Thus, we obtain from [2, p. 578] that the equation (6.5) has a unique solution S2 which is given by formula (6.6). This matrix S2 is positive Hermitian. We introduce the block diagonal matrix R := diag (ρ, ρ, . . . , ρ) n

where ρ is a positive Hermitian m × m matrix. Let us consider the equation S1 − AS1 A∗ = Ψ1 Ψ∗1

(6.7)

where the complex mn × mn matrix Ψ1 is deﬁned by the relations Ψ1 := R−1 · col(a1 , a2 , . . . , aN ) and ak = col(a1,k , a2,k , . . . , ank ,k ),

k ∈ {1, . . . , N }.

Here the matrices ak and aj,k have the sizes mnk × m and m × m, respectively. Using [2, Theorem A3.4, part (a)] we obtain the following result.

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Proposition 6.2. There is one and only complex mn × mn matrix S1 satisfying (6.7), namely ∞ Aj Ψ1 Ψ∗1 (Aj )∗ . S1 = j=0

Moreover, the matrix S1 is nonnegative Hermitian. The identities (6.5) and (6.7) generate an extremal interpolation problem to which the results of Section 1–3 can be applied. We omit here its exact formulation.This will be done in detail in a forthcoming paper.

References [1] Akhiezer, N.I.: On a minimum problem in function theory and the number of roots of an algebraic equation inside the unit disc (Russian), Izv. Akad. Nauk SSR, Otdel. Mat. i Est. Nauk 9 (1930), 1169–1189, English Translation in: Topics in Interpolation Theory (Eds.: H. Dym, B. Fritzsche, V.E. Katsnelson, B. Kirstein), OT Series, Volume 95, Birkh¨ auser, Basel-Boston-Berlin 1997, pp. 19–35. [2] Ball, J.A.; Gohberg, I.; Rodman, L.: Interpolation of Rational Matrix Functions, OT Series, Volume 45, Birkh¨auser, Basel-Boston-Berlin 1990. [3] Ferrante, A.; Levy, B.C.: Hermitian solutions of the equation X = Q + N X −1 N ∗ , Linear Alg. Appl. 247 (1996), 359–373. [4] Helton, J.W.; Sakhnovich, L.A.: Extremal problems of interpolation theory, Rocky Mount. J. Math. 35 (2005), 819–841. [5] Kimura, H.: State space approach to the classical interpolation problem and its applications, in: Three Decades of Mathematical System Theory (Eds.: H. Nijmeijer, J.M. Schumacher), Lect. Notes Contr. Int. Sci., Volume 135, Springer, Berlin 1989, pp. 243–275. [6] Ran, A.C.M.; Reurings, M.C.B.: A nonlinear matrix equation connected to interpolation theory, Linear Alg. Appl. 379 (2004), 289–302. [7] Sakhnovich, L.A.: Interpolation Theory and Its Applications, Kluwer, Dordrecht 1997. Bernd Fritzsche and Bernd Kirstein Fakult¨ at f¨ ur Mathematik und Informatik Universit¨ at Leipzig Postfach: 10 09 20 D-04009 Leipzig, Germany e-mail: [email protected] [email protected] Lev A. Sakhnovich 99 Cove Avenue Milford, CT 06461, USA e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 199, 347–358 c 2010 Birkh¨ auser Verlag Basel/Switzerland

O(n) Algorithms for Banded Plus Semiseparable Matrices Jitesh Jain, Hong Li, Cheng-Kok Koh and Venkataramanan Balakrishnan Abstract. We present a new representation for the inverse of a matrix that is a sum of a banded matrix and a semiseparable matrix. In particular, we show that under certain conditions, the inverse of a banded plus semiseparable matrix can also be expressed as a banded plus semiseparable matrix. Using this result, we devise a fast algorithm for the solution of linear systems of equations involving such matrices. Numerical results show that the new algorithm competes favorably with existing techniques in terms of computational time. Mathematics Subject Classiﬁcation (2000). 15A09, 15A23, 65F05, 65L10, 65R20. Keywords. Semiseperable matrix, fast algorithms, linear solver, inverse, structured matrices.

1. Introduction Understanding of structured matrices and computation with them have long been problems of theoretical and practical interest [1]. Recently a class of matrices called semiseparable matrices has received considerable attention [2, 3, 4, 5, 6, 7, 8, 9]. Perhaps the simplest example of a semiseparable matrix is given by the inverse of a symmetric tridiagonal matrix: If A = AT is irreducible tridiagonal and nonsingular, then it is well known that A−1 can be written as: 2 ui vj if i ≤ j, −1 Aij = (1) uj vi if i > j. Matrices such as the one in (1) and its generalizations arise in a number of practical applications like integral equations [10, 11], statistics [12], and vibrational analysis [13]. Modeling with a semiseparable matrix evidently oﬀers the potential of reducing the number of parameters describing a matrix by up to an order of magnitude (from O(n2 ) to O(n) in the example in (1)). Moreover, it is known that This work was done when the authors were at Purdue University.

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computation with semiseparable matrices requires signiﬁcantly reduced eﬀort; for example, for a semiseparable matrix of the form in (1), matrix-vector multiplies can be performed in O(n) (as compared to O(n2 ) in general). In several practical situations, semiseparable matrices do not arise alone; instead, matrices that are encountered, are a sum of diagonal and a semiseparable matrix or a banded and a semiseparable matrix. Examples where such matrices arise, are in boundary value problems [14, 15], and integral equations [16]. The computation with such matrices has been a subject of considerable interest. Several algorithms have been developed to deal with matrix inversion and linear equation solution with such matrices. Formulae for inversion of diagonal plus semiseparable matrices were ﬁrst developed in [6]. However, these formulae were valid under the assumption that the matrix is strongly regular, i.e., it has non-vanishing principal minors. These restrictions were later removed in [4]. Recently several algorithms have been developed for solving linear systems of equation Ax = c,

(2)

where the coeﬃcient matrix A is a sum of diagonal and semiseparable matrix [3, 4, 5, 7]. Fast and numerically stable algorithms for banded plus semiseparable linear system of equations were proposed in [2]. We present two main results in this paper. First, we provide an explicit representation for inverses of banded plus semiseparable matrices. In particular, we show that under certain conditions, the inverse of a banded plus semiseparable matrix is again a banded plus semiseparable matrix. Our second contribution is to provide fast solutions of linear systems of equations with these matrices. A comparison with the state of the art shows that our method is about two times faster than existing solutions of linear system with diagonal plus semiseparable matrices. When banded plus semiseparable matrices are considered, our method is up to twenty times faster than existing solutions. The remainder of the paper is organized as follows. In §2, we establish mathematical preliminaries and the notation used in the paper, as well as a brief review of the state of the art. In §3 we present formulae for the inverse of banded plus semiseparable matrices. We exploit this result in §4 to provide a fast algorithm for solving linear systems of equations. In §5, we establish the eﬀectiveness of the new algorithm via numerical results. The extension of this algorithm to handle some special cases is presented in an appendix.

2. Preliminaries n

n

For k = 1, . . . , a, and r = 1, . . . , b, let uk = {uk (i)}i=1 , vk = {vk (i)}i=1 , pr = {pr (i)}ni=1 , and qr = {qr (i)}ni=1 , be speciﬁed vectors. Then Sab , an n × n semiseparable matrix of order (a, b), is characterized as follows: 2 a uk (i)vk (j) if i ≤ j, k=1 Sij = (3) b p if i > j. r=1 r (j)qr (i)

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We use Sn as the generic notation for the class of semiseparable matrices of size n. We use Bn to denote the class of banded matrices of size n. Blm = {Bij }ni,j=1 is used to denote a banded matrix with l non-zero diagonals strictly above the main diagonal and m non-zero diagonals strictly below the main diagonal, i.e., if Blm ∈ Bn then Bij = 0 if i − j > m or j − i > l. The numbers l and m are called respectively the upper and lower bandwidths of a banded matrix Blm . Dn is used to denote the class of diagonal matrices. D = diag(d) is used to denote a diagonal matrix with d as the main diagonal. We now deﬁne a proper banded matrix. Deﬁnition 2.1. A nonsingular banded matrix Blm is said to be proper if any submatrix obtained by deleting r(= max(l, m)) consecutive rows and r consecutive columns is nonsingular. It is well known that the inverses of banded matrices are semiseparable matrices. Theorem 2.2 ([17]). Let Blm be a n × n proper banded matrix. Then its inverse can be written as −1 (Blm ) = Slm , Slm ∈ Sn . Remark 2.3. The above semiseparable representation, though elegant and compact, suﬀers from numerical instabilities [18], making it of limited practical use. Hence, the above representation will be only used as a theoretical tool, and not in any numerical implementations. We next present a brief review of the state of the art for solving linear system of equation in (2), where A is a sum of banded and semiseparable matrices. We ﬁrst consider the case when A is a sum of a diagonal and a semiseparable matrix. Two algorithms for solving such systems were developed in [3]. The ﬁrst step is the same with both algorithms, where A ∈ n × n is reduced to a upper Hessenberg matrix H via n − 1 Givens rotations: A = GT2 GT3 · · · GTn H. GT 2,...,n

It was shown that GT2,...,n is a lower Hessenberg matrix, whose upper triangular part is the upper triangular part of a unit-rank matrix. The upper triangular part of H was shown to be the upper triangular part of a matrix of rank two. The second step of both algorithms is to reduce the upper Hessenberg matrix H into an upper triangular matrix via n − 1 Givens rotations. Two diﬀerent algorithms were obtained by either applying the Givens rotations on the left of H, leading to QR algorithm, or applying Givens rotations to the right of H, obtaining the U RV algorithm. Exploiting the low rank structure of GT2,...,n and H, both algorithms were shown to require 54n − 44 ﬂops as compared to 58n ﬂops for the algorithm in [5] and 59n ﬂops for the one in [2].

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The authors in [2] proposed a fast and numerical stable algorithm for the more general case of the solution of (2) when A is a sum of banded and semiseparable matrix. The basic idea is to compute a two-sided decomposition of the matrix A such that A = W LH. Here L is a lower triangular matrix, and both W and H can be written as a product of elementary matrices. An eﬃcient algorithm for solving (2) is obtained by inverting W , L, and H on the ﬂy. The matrices W and H can be either obtained via standard Gaussian elimination, or by using Givens rotations and Householder reﬂection matrices. The algorithm based on Gaussian elimination was shown to be marginally better in computational time than the one based on Givens rotations and Householder reﬂections, with the latter algorithm performing better with respect to accuracy. For a banded matrix with upper and lower bandwidth l and m and a semiseparable matrix of size n and order (a, b), the algorithm based on Givens rotation and Householder reﬂection was shown to have an operation 5a) (l + a) n ﬂops, while the algorithm based count of 11a2 + 2 (2l + 2m + 3b + on Gaussian elimination requires 9a2 + 2 (l + 2m + 2b + 2a) (l + a) n ﬂops. The approach we take in this paper is to ﬁrst provide an explicit representation for inverses of banded plus semiseparable matrices. We then exploit these results to come up with a fast algorithm for solving linear systems of equations.

3. Structure for inverses of banded plus semiseparable matrices We begin this section by considering semiseparable matrices of order (1,1). We ﬁrst present a theorem on multiplicative structure of inverses of banded plus semiseparable matrices of order (1, 1). Theorem 3.1. Let Blm be a n × n banded matrix and S11 be a n × n semiseparable matrix of order (1, 1). Then the inverse of their sum has the following multiplicative structure: −1 −1 m ˜ Bl + S11 = DLT B m+1 LD, l+1

˜ ∈ Dn , B m+1 ∈ Bn , and L is a lower bidiagonal matrix. where D, D l+1 Proof: Let

2

ui vj if i ≤ j, pj qi if i > j. Assume v1 (i) = 0, q1 (i) = 0 for this and the next section. The results have been extended for the more general case in the appendix. Let Dv = diag(1/v) and Dq = diag(1/q) be diagonal matrices. Let L be a lower bidiagonal matrix, which is deﬁned as follows: ⎞ ⎛ 1 ⎟ ⎜−1 1 ⎟ ⎜ ⎟ ⎜ −1 1 (4) L=⎜ ⎟. ⎟ ⎜ .. .. ⎠ ⎝ . . S11

ij

=

−1 1

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It can be easily veriﬁed that ˆ LDq S11 Dv LT = L,

(5)

Dv LT Sˆ11 LDq = S˜11 + D,

(6)

˜ LDq Sˆ11 Dv LT = S˜11 + L,

(7) 1 ˜1 ˆ ˜ ˆ where L, L are lower bidiagonal matrices, D ∈ Dn , and S1 , S1 ∈ Sn . Now, from (5) ˆ LDq Blm + S11 Dv LT = LDq Blm Dv LT + L. It can be readily veriﬁed that matrix LDq Blm Dv LT is banded with lower and ˜ m+1 ∈ Bn . upper bandwidth l + 1 and m + 1 respectively, i.e., LDv Blm DqT LT = B l+1 ˆ is lower bidiagonal, we have Moreover, as L m+1 LDq Blm + S11 Dv LT = Bl+1 Thus

Blm + S11

−1

m+1 −1 = Dv LT Bl+1 LDq .

m+1 in Theorem 3.1 is proper. Then Remark 3.2. Suppose the banded matrix Bl+1 using Theorem 2.2 and (6), we have the following elegant additive structure for the inverse of a banded and semiseparable matrix of order (1, 1): m −1 m+1 Bl + S11 = Dv LT S˜l+1 LDq m+1 = Sl+1 + D, m+1 ∈ Sn , and D ∈ Dn . where Sl+1

More generally, the following theorem characterizes the inverses of general banded plus semiseparable matrices. Theorem 3.3. Let Blm be a n × n banded matrix and Sab be a n × n semiseparable matrix. Then the inverse of their sum has the following multiplicative structure: m −1 m+b −1 Bl + Sab = D1 LT · · · Db LT Bl+a LDa · · · LD1 , m+b where D1 , D1 , D2 , D2 , . . . , Da , Da , Db , Db ∈ Dn , Bl+a ∈ Bn , and L is a lower bidiagonal matrix.

Proof: Without loss of generality, assume b ≥ a. Now, from (5) and (7) m+1 b−1 LD1 Blm + Sab D1 LT = Bl+1 + Sa−1 m m+a LDa · · · LD1 Bl + Sab D1 LT · · · Da LT = Bl+a + S0b−a m+b LDa · · · LD1 Blm + Sab D1 LT · · · Da LT · · · Db LT = Bl+a , where D1 , D1 , D2 , D2 , . . . , Db , Db ∈ Dn . Thus m −1 m+b −1 Bl + Sab = D1 LT · · · Da LT · · · Db LT Bl+a LDa · · · LD1 .

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m+b in Theorem 3.3 is proper. Then Remark 3.4. Suppose the banded matrix Bl+a using Theorem 2.2 and (6), we have the following elegant additive structure for the inverse of a banded and semiseparable matrix: m −1 m+b Bl + Sab = D1 LT · · · Da LT · · · Db LT S˜l+a LDa · · · LD1 b−1 m+b = Ba−1 + Sl+a , b−1 m+b ∈ Bn , and Sl+a ∈ Sn . where Ba−1

Corollary 3.5. The inverse of a diagonal plus semiseparable matrix is again a diagonal plus semiseparable matrix.

4. Fast solution of Ax = c In this section, we consider the problem of ﬁnding the solution of linear systems of equation, Ax = c, where the coeﬃcient matrix, A is a sum of banded and semiseparable matrix, i.e., A = Blm + Sab , Blm ∈ Bn , Sab ∈ Sn . Without loss of generality assume b ≥ a. −1 m+b −1 x = A−1 c = Blm + Sab c = D1 LT · · · Da LT · · · Db LT Bl+a LDa · · · LD1 c. (8) function x = Ainvc(Sab , Blm , c) m+b 1. Calculate D1 , D1 , . . . , Da , Da , . . . , Db , Db , and Bl+a as described in proof of Theorem 3.3; 2. z = LDa · · · LD1 c ; m+b m+b 3. y = Bl+a \ z ( Solve Bl+a y = z ); 4. x = D1 LT · · · Da LT · · · Db LT y; return x; Note that for solving Ax = c, we do not need the condition of the banded matrices being proper. We now present a complexity analysis of the above-mentioned procedure of solving Ax = c. We will assume 1 " a, b, l, m " n to make the complexity analysis simpler. The ﬂop count of the overall algorithm is dominated by the cost of steps 1 and 3. Total cost of step-1, i.e., calculating D1 , D1 , . . . , Da , Da , . . . , Db , Db , m+b and Bl+a is 2 (l + m) max (a, b) + 8 a2 + b2 n ﬂops. In step-3 the cost of solving a banded system of equations is 2(l + a)(m + b)n ﬂops. Hence the total complexity of the proposed algorithm is 2(l + a)(m + b) + 2 (l + m) max (a, b) + 8 a2 + b2 n operations. For diagonal plus semiseparable systems, the complexity reduces to 30n operations.

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5. Numerical results In this section, we compare the results for computational time and accuracy for solving a linear system of equations of form Ax = b, for several algorithms. We compare the results for following four algorithms: • Algorithm I: QR algorithm for solving a diagonal plus semiseparable system, as given in [3]. • Algorithm II: U RV algorithm for solving a diagonal plus semiseparable system, as given in [3]. • Algorithm III: Chandrasekaran-Gu algorithm for banded plus semiseparable system of equations, as given in [2]. • Algorithm IV: The new algorithm as described in §4. All numerical experiments were performed in MATLAB running on a 4 CPU r machine. For Algorithm I and II, we used the author’s r Pentium# 1.5GHz Intel# implementation, taken directly from [19]. All the matrix entries are randomly generated, drawn from a Gaussian distribution with zero mean and unit variance. 5.1. A is a sum of diagonal and semiseparable matrix We ﬁrst present the computational requirements and the accuracy of our approach against Algorithm I and II from [3]. The matrix A comprises of a diagonal plus a semiseparable matrix. Table 1 summarizes the results. Error in solving x in ∞ Ax = b is deﬁned as (Ax−b) . As expected all three algorithms are linear in A∞ x∞ computational time. Algorithm IV is faster than and comparable in accuracy to Algorithms I and II. For a system with size 320000, Algorithm IV is 1.9× faster than Algorithm I and 2.4× faster than Algorithm II. This supports the theoretical complexities mentioned previously, where Algorithm IV takes 30n operations as compared to 54n − 44 operations taken by Algorithm I and II. Size

10000 20000 40000 80000 160000 320000 640000 1280000

Error =

(Ax−b)∞ A∞ x∞

Time (in sec)

Alg I [3]

Alg II [3]

Alg IV

3.33 × 10−19 8.01 × 10−19 9.24 × 10−19 1.47 × 10−18 1.29 × 10−19 2.34 × 10−19 6.53 × 10−20 8.30 × 10−20

5.32 × 10−19 4.66 × 10−19 1.80 × 10−18 8.27 × 10−19 5.49 × 10−19 5.24 × 10−19 1.71 × 10−19 8.40 × 10−19

3.27 × 10−18 1.40 × 10−17 5.63 × 10−18 2.15 × 10−17 5.65 × 10−19 2.30 × 10−18 8.91 × 10−18 2.88 × 10−18

Alg I [3] Alg II [3] Alg IV .34 .68 1.39 2.77 5.59 11.06 22.02 44.17

.43 .85 1.74 3.50 7.00 13.92 27.79 55.89

.14 .27 .65 1.41 3.05 5.86 11.86 24.39

Table 1. Error values and Computational time as compared with Algorithms I and II.

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5.2. A is a sum of banded and semiseparable matrix We now present the computational requirements and the accuracy of our approach against Algorithm III from [2]. The matrix A comprises of a banded plus semiseparable matrix. The three variables of interest of the coeﬃcient matrix are the bandwidth of the banded matrix, order of the semiseparable matrix and size of the system. We ﬁrst give results by varying one of the quantities at a time, keeping other two constant. Table 2 shows the results for increasing sizes of the linear Size

1000 2000 3000 4000 5000 6000 7000 8000

Error =

(Ax−b)∞ A∞ x∞

Time (in sec)

Alg III [2]

Alg IV

Alg III [2]

Alg IV

8.58 × 10−17 6.41 × 10−17 1.78 × 10−16 7.14 × 10−17 1.84 × 10−16 1.20 × 10−16 9.05 × 10−17 3.73 × 10−17

3.07 × 10−13 8.42 × 10−13 2.68 × 10−13 1.57 × 10−12 1.87 × 10−12 1.25 × 10−12 8.48 × 10−13 1.03 × 10−12

1.33 2.65 3.96 5.29 6.57 7.98 9.27 10.62

0.13 0.26 0.39 0.53 0.65 0.80 0.96 1.12

Table 2. Error values and Computational time as compared with Algorithm III; l = m = a = b = 5.

systems. The upper and lower bandwidth of the banded matrix in all cases is 5. The order of the semiseparable matrix is also kept constant at (5, 5). Algorithm IV performs favorably in terms of computational time, being 10× faster than Algorithm IV for the size of 8000. Similar results are seen in Table 3 and 4, where we are varying the bandwidth and the order respectively. Algorithm III exhibits better accuracy than Algorithm IV, as it relies heavily on Givens rotations. We now give results when all three variables are varied at the same time. Table 5 shows the results for increasing sizes of the linear systems. The upper and lower bandwidth of banded matrices as well as the order of semiseparable matrices n are varied as 250 , where n denotes the size of the system. Algorithm IV is faster, but at the expense of numerical accuracy. The computational times are consistent with the theoretical ﬂop count. For a = b = l = m = r, ﬂop count of Algorithm III reduces to 59r2 n. For the same case, ﬂop count of the proposed Algorithm is 28r2 n.

Algorithms for Banded Plus Semiseparable Matrices l=m

Error = Alg III [2]

11 21 31 41 51 61 71 81

−17

7.51 × 10 2.33 × 10−16 8.64 × 10−16 6.88 × 10−16 4.32 × 10−16 1.33 × 10−15 4.95 × 10−16 9.71 × 10−16

(Ax−b)∞ A∞ x∞

355

Time (in sec)

Alg IV −13

3.34 × 10 2.40 × 10−13 4.82 × 10−13 1.22 × 10−12 5.61 × 10−13 2.33 × 10−12 1.26 × 10−12 2.87 × 10−12

Alg III [2]

Alg IV

2.24 3.57 4.90 5.54 5.60 7.24 8.19 9.61

0.16 0.24 0.35 .46 .55 .67 .71 .81

Table 3. Error values and Computational time as compared with Algorithm III; n = 4000, a = b = 1.

a=b

Error = Alg III [2]

11 21 31 41 51 61 71 81

−16

2.41 × 10 3.52 × 10−16 9.79 × 10−16 6.37 × 10−16 1.16 × 10−15 1.01 × 10−15 1.68 × 10−15 1.70 × 10−15

(Ax−b)∞ A∞ x∞

Time (in sec)

Alg IV −10

2.70 × 10 2.88 × 10−10 1.82 × 10−8 2.48 × 10−9 3.75 × 10−9 6.06 × 10−9 1.62 × 10−7 1.03 × 10−7

Alg III [2] Alg IV 8.96 15.47 25.29 43.19 61.07 117.37 188.40 288.77

1.09 3.20 6.04 9.40 14.83 21.32 27.26 34.94

Table 4. Error values and Computational time as compared with Algorithm III; n = 4000, l = m = 1.

6. Conclusions We have presented a representation for inverse of banded plus semiseparable matrices. We have also presented fast algorithms for solving linear system of equations with these matrices. Numerical results show that the proposed approach competes favorably with the state of the art algorithms in terms of computational eﬃciency.

Appendix In the discussion till now, we have assumed we are given a semiseparable matrix Sab , as deﬁned in (3), such that for all i, k vk (i) = 0, qk (i) = 0. Now, we will give procedure to modify the proposed methods when such assumptions do not hold

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Size

1000 2000 3000 4000 5000 6000 7000 8000

Error =

(Ax−b)∞ A∞ x∞

Time (in sec)

Alg III [2]

Alg IV

Alg III [2]

Alg IV

7.00 × 10−17 6.68 × 10−17 2.69 × 10−16 2.51 × 10−16 5.14 × 10−16 2.32 × 10−16 2.55 × 10−16 6.37 × 10−16

1.21 × 10−12 7.69 × 10−12 1.69 × 10−11 1.46 × 10−9 6.25 × 10−10 1.11 × 10−10 2.67 × 10−9 3.39 × 10−10

1.31 4.49 8.84 16.46 26.36 41.16 63.70 96.16

0.13 0.90 2.71 6.53 11.62 19.50 30.93 45.39

Table 5. Error values and Computational time as compared with Aln gorithm III; l = m = a = b = 250 .

true. We will only consider the symmetric case, i.e., p = u, q = v. In addition, we assume that the order of semiseparable matrices in consideration is (1, 1), i.e., l = m = 1. Hence 2 ui vj if i ≤ j, Sij = uj vi if i > j. The general case can be handled in a similar fashion. Let us assume vk = 0, and vi = 0 if i = k. The technique that we propose next can be easily extended to handle the case when for more than one i, vi is zero. Consider the matrices L, and Dv as deﬁned in §3. In addition, let L(k, k − 1) = 0, and Dv (k, k) = 1. Then it can be easily veriﬁed that LDv S11 Dv LT = Mu,v , where M is deﬁned as follows: ⎛ α1 ⎜ α2 ⎜ ⎜ ⎜ ⎜ Mu,v = ⎜ ⎜ ⎜α1 α2 ⎜ ⎜ ⎝

..

.

0 . . . αk

⎞

α1 α2 .. .

⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎠

αk αk+1 ..

. αn

/ where α1 = u1 v1 , αk = uk , αk+1 = uk+1 − uk , αi = ui vi − ui−1 vi−1 for all i ∈ {1, k, k + 1}. We can now do a tridiagonal decomposition of Mu,v as P M P T = T,

Algorithms for Banded Plus Semiseparable Matrices where T is a tridiagonal matrix, and P is given by ⎛ 1 1 −α α2 2 ⎜ 1 −α α3 ⎜ ⎜ .. .. P =⎜ . . ⎜ ⎝ 1 − ααn−1 n 1

357

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Proceeding as with the proof of Theorem 1, we can show that m −1 m+2 −1 ˜ Bl + S11 = DLT Bl+2 LD, ˜ ∈ Dn , B m+2 ∈ Bn , and L is a lower bidiagonal matrix. where D, D l+2

References [1] T. Kailath and A.H. Sayed. Fast reliable algorithms for matrices with structure. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1999. [2] S. Chandrasekaran and M. Gu. Fast and Stable algorithms for banded plus semiseparable systems of linear equations. SIAM Journal on Matrix Analysis and Applications, 25(2):373–384, 2003. [3] E. Van Camp, N. Mastronardi and M. Van Barel. Two fast algorithms for solving diagonal-plus-semiseparable linear systems. Journal of Computational and Applied Mathematics, 164-165:731–747, 2004. [4] Y. Eidelman and I. Gohberg. Inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices. Computer and Mathematics with Applications, 33(4):69–79, 1996. [5] Y. Eidelman and I. Gohberg. A modiﬁcation of Dewilde-van der Veen method for inversion of ﬁnite structured matrices. Linear Algebra Appl., 343-344:419–450, 2002. [6] I. Gohberg and T. Kailath and I. Koltracht. Linear Complexity algorithms for semiseparable matrices. Integral Equations and Operator Theory, 8:780–804, 1985. [7] N. Mastronardi, S. Chandrasekaran and S.V. Huﬀel. Fast and stable two-way algorithm for diagonal plus semi-separable system of linear equations. Numerical Linear Algebra Appl., 8(1):7–12, 2001. [8] P. Rozsa. Band Matrices and semi-separable matrices. Colloquia Mathematica Societatis Janos Bolyai, 50:229–237, 1986. [9] P. Rozsa, R. Bevilacqua, F. Romani and P. Favati. On Band Matrices and their inverses. Linear Algebra Appl., 150:287–295, 1991. [10] E. Asplund. Inverses of Matrices aij which satisfy aij = 0 for j > i + p. Mathematica Scandinavica, 7:57–60, 1959. [11] S.O. Asplund. Finite boundary value problems solved by Green’s Matrix. Mathematica Scandinavica, 7:49–56, 1959. [12] S.N. Roy and A.E Sarhan. On inverting a class of patterned matrices. Biometrika, 43:227–231, 1956. [13] F.R. Gantmacher and M.G. Krein. Oscillation matrices and kernels and small vibrations of mechanical systems. AMS Chelsea publishing, 2002.

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[14] L. Greengard and V. Rokhlin. On the numerical solution of two-point boundary value problems. Communications on Pure and Applied Mathematics, 44:419–452, 1991. [15] J. Lee and L. Greengard. A fast adaptive numerical method for stiﬀ two-point boundary value problems. SIAM Journal on Scientiﬁc Computing, 18:403–429, 1997. [16] Starr, Jr., Harold Page. On the Numerical Solution of One-Dimensional Integral and Diﬀerential Equations. Department of Computer Science, Yale University, New Haven, CT, 1992. [17] F. Romani. On the additive structure of inverses of banded matrices. Linear Algebra Appl., 80:131–140, 1986. [18] P. Concus and G. Meurant. On Computing INV block preconditionings for the conjugate gradient method. BIT, 26:493–504, 1986. [19] http://www.cs.kuleuven.ac.be/∼marc/software/index.html. Jitesh Jain Intel Corporation Hillsboro, OR 97124, USA e-mail: [email protected] Hong Li Synopsys Inc. Mountain View, CA 94043, USA e-mail: [email protected] Cheng-Kok Koh and Venkataramanan Balakrishnan School of Electrical and Computer Engineering Purdue University West Lafayette, IN 47907-1285, USA e-mail: [email protected] [email protected]

Operator Theory: Advances and Applications, Vol. 199, 359–375 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Uniﬁed Nearly Optimal Algorithms for Structured Integer Matrices Victor Y. Pan, Brian J. Murphy and Rhys Eric Rosholt Abstract. Our subject is the solution of a structured linear system of equations, which is closely linked to computing a shortest displacement generator for the inverse of its structured coeﬃcient matrix. We consider integer matrices with the displacement structure of Toeplitz, Hankel, Vandermonde, and Cauchy types and combine the uniﬁed divide-and-conquer MBA algorithm (due to Morf 1974, 1980 and Bitmead and Anderson 1980) with the Chinese remainder algorithm to solve both computational problems within nearly optimal randomized Boolean and word time bounds. The bounds cover the cost of both solution and its correctness veriﬁcation. The algorithms and nearly optimal time bounds are extended to the computation of the determinant of a structured integer matrix, its rank and a basis for its null space and further to some fundamental computations with univariate polynomials that have integer coeﬃcients. Mathematics Subject Classiﬁcation (2000). 68W30, 68W20, 68Q25, 68W40. Keywords. Structured matrices, the MBA divide-and-conquer algorithm.

1. Introduction Linear systems of equations with displacement structure of Toeplitz, Hankel, Vandermonde and Cauchy types are omnipresent in scientiﬁc and engineering computations and signal and image processing. Due to the structure they can be solved fast, in quadratic rather than cubic arithmetic time [L47], [D59], [T64] or even Some results of this paper have been presented at the Annual International Conference on Application of Computer Algebra, Volos, Greece, June 2002; ACM International Symposium on Symbolic and Algebraic Computation, Lille, France, July 2002; and the 5th Annual Conference on Computer Algebra in Scientiﬁc Computing, Yalta, Crimea, Ukraine, September 2002. Supported by NSF Grant CCR 9732206 and PSC CUNY Awards 67297–0036, 68291–0037, 69330–0038, and 69350–0038.

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superfast, in nearly linear time [BGY80], [M80], [BA80], [OP98], [P01]. Quite typically, however, application of the superfast algorithms leads to the problems of numerical stability [B85]. Moreover, structured linear systems of some important classes are ill conditioned [GI88], [T94]. This suggests devising fast and superfast symbolic algorithms, applied to the case of integer input coeﬃcients, to which one can reduce the rational inputs by scaling. The complexity of such algorithms is usually analyzed under the Boolean (bit-operation) and word operation models [GG03]. Our main goal in this paper is to present the algorithm in a uniﬁed way for the general class of integer input matrices with displacement structure and to estimate the complexity of this algorithm and the probability of degeneration (cf. [OP98], [P01], [PW08]). Besides the complexity and degeneration issues, we elaborate upon recursive compression of the displacement generators in the MBA process, displacement transformation of the input matrices, and reconstruction of the rational solution from its representation modulo selected distinct random primes. We also cover the extension to computations with structured singular matrices. We arrive at the complexity bounds that cover both randomized solution and its correction veriﬁcation and that are nearly optimal (up to a polylogarithmic factor) under both Boolean and word operation models. The bounds cover the cost of both solution and its correctness veriﬁcation. Versus the information lower bound of n2 log n, the algorithm takes the order of n2 log3 n bit-operations for a structured linear system of n equations with n unknowns where all coeﬃcients have absolute values in nO(1) . The same cost bound covers the computation of the determinant, rank, and a basis for the null space. The algorithm can be called superfast because its Boolean cost bound is nearly quadratic versus the orders of n4 and n3 bit-operations required for the solution by Gaussian elimination and by the fast algorithms such as Levinson–Durbin’s and Trench’s, respectively. Our nearly optimal complexity estimates can be extended to Berlekamp–Massey’s reconstruction of a linear recurrence coeﬃcients from its values and to computing the greatest common divisor, least common multiples, and Pad´e approximation for univariate polynomials. We organize our presentation as follows. In the next section we state some deﬁnitions and basic results. We describe the uniﬁed MBA divide-and-conquer algorithm in Section 3. In Section 4 we cover the reconstruction of the rational solution from the solution modulo suﬃciently many distinct primes and estimate the overall Boolean and word complexity of our computations. In Section 5 we recall various related works. The presented algorithms have been implemented by the third and mostly the second authors. Otherwise the paper is due to the ﬁrst author.

2. Deﬁnitions and basic facts 2.1. Integers We write log for log2 (unless we specify otherwise), Z for the ring of integers, Zq for the ring of integers modulo an integer q, and Q for the ﬁeld of rational numbers.

Uniﬁed Nearly Optimal Algorithms

361

˜ (n)) denotes O(f (n)(log log n)c ) for “Ops” stand for “arithmetic operations”. O(f a constant c. f (s) = O(1) as well as “f (s) is in O(1)” means that f (s) is bounded by a constant independent of s. We write a = z mod q, for three integers q > 1, a, and z, either to denote a unique integer a such that q divides z − a and 0 ≤ a < q or, wherever due to the context this causes no confusion, just to show that q divides z − a. Fact 2.1. Assume two integers a and m such that m > 1 and m > 2|a|. Then the following expressions hold, a=a

mod m if 2|a mod m| < m,

a=a

mod m − m otherwise.

2.2. Polynomial and integer multiplication Deﬁnition 2.1. Let μ(d)) denote the minimum number of bit operations suﬃcient to perform an arithmetic operation modulo a prime q < 2d , including division by an integer coprime with q, and let m(n) denote the minimum number of ﬁeld operations suﬃcient to multiply two polynomials of degree n − 1 or less over any ﬁeld, ring with unity, or algebra. Fact 2.2. We have μ(d) = O((d log d) log log d) and m(n) = O((n log n) log log n) [CK91], [K98], [B03], [GG03], [F07]. 2.3. General matrices k×l Deﬁnition 2.2. M = (mi,j )k,l is a k × l matrix with entries mi,j in a ring i,j=1 ∈ R k k×1 is a column vector. I is the identity matrix of a proper size. R. v = (vi )i=1 ∈ R Il is the l × l identity matrix. (K, L) is a 1 × 2 block matrix with the blocks K and L. D(v) = diag(v) = diag(vi )i is the diagonal matrix with the diagonal entries dii = vi given by the coordinates of the vector v = (vi )i . M T is the transpose of M . M (h) is the h×h leading principal (that is northwestern) submatrix of M. A matrix M of rank ρ has generic rank proﬁle if its submatrices M (k) are nonsingular for k = 1, . . . , ρ, that is up to the rank size ρ × ρ. M is strongly nonsingular if it is nonsingular and has generic rank proﬁle. A block of a matrix is its submatrix in the intersection of a set of its contiguous rows and a set of its contiguous columns. Deﬁnition 2.3. The symmetric matrix M T M for a nonsingular matrix M is called positive deﬁnite. (In Qn×n such a matrix is strongly nonsingular.) Deﬁnition 2.4. det M and adj M are the determinant and the adjoint of a matrix M, respectively. (adj M = M −1 det M if M is nonsingular.) Deﬁnition 2.5. |M | = ||M ||∞ = maxi j |mi,j | is the row norm of a matrix M = (mi,j )i,j ; α(M ) = maxi,j |mi,j |; |M |/n ≤ α(M ) ≤ |M |; |v| = β(v) = maxi |vi | is the maximum norm of a vector v = (vi )i . Deﬁnition 2.6. mS ≤ 2n2 − n is the minimum number of arithmetic operations in an algorithm that multiplies an n × n matrix S by a vector.

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Clearly, we can multiply a pair of n × n matrices by using 2n3 − n2 arithmetic operations. We refer the reader to [P84], [CW90], [K04], and the bibliography therein on theoretical and practical speed up. √ 1/2 Fact 2.3. | det M | ≤ j (Σi m2i,j ) ≤ (α(M ) n)n , and so (since the entries of the matrix adj M are the√determinants of (n − 1) × (n − 1) submatrices of M ), we have α(adj M ) ≤ (α(M ) n − 1)n−1 for an n × n matrix M = (mi,j )i,j . √ Hadamard’s bound | det M | ≤ (α(M ) n)n above is known to be sharp in the worst case, but is an over-estimate on the average according to [ABM99]. Remark 2.1. We can apply Fact 2.1 and readily recover the integer det √ M as soon as we √ have its value computed modulo p+ for some p+ > 2(α(M ) n)n because (α(M ) n)n ≥ | det M | in virtue of Fact 2.3. If M is a nonsingular matrix in Zp+ and if its determinant det M and inverse M −1 have been computed in Zp+ , then we can immediately compute in Zp+ the matrix adj M = M −1 det M . In virtue of Fact 2.3, its integer entries lie in the range (−p+ /2, p+/2), and we can recover them from their values modulo p+ by applying Fact 2.1 again. 2.4. Matrices with displacement structure: general properties In this subsection we deﬁne matrices with displacement structure and recall their basic properties. Deﬁnition 2.7. ΔA,B (M ) = M − AM B = GH T (resp. ∇A,B (M ) = AM − M B = GH T ) is the Stein (resp. Sylvester) displacement of an n × n matrix M where n × n matrices A and B are operator matrices and a pair of n × l matrices G and H form a displacement generator of length l for the matrix M . The rank of the displacement is called the displacement rank of the matrix M . (It minimizes the length l of displacement generators ΔA,B (M ) (resp. ∇A,B (M )) for a ﬁxed triple (A, B, M ).) The simple basic results below are from [P01, Theorems 1.3.1, 1.5.1–1.5.6]. Theorem 2.1. If the matrix A (resp. B) is nonsingular, then we have ΔA,B (M ) = A−1 ∇A−1 ,B (M ) (resp. ΔA,B (M ) = −∇A,B −1 (M )B −1 ). Theorem 2.2. For matrices A, B, M , and N of compatible sizes, displacement operators L = ΔA,B and L = ∇A,B , and a scalar a, we have L(M + aN ) = L(M ) + aL(N ), ΔA,B (M T ) = (ΔB T ,AT (M ))T , ∇A,B (M T ) = −(∇B T ,AT (M ))T . Furthermore ∇B,A (M −1 ) = −M −1 ∇A,B (M )M −1 if M is a nonsingular matrix, ΔB,A (M −1 ) = BM −1 ΔA,B (M )B −1 M −1 if the matrices B and M are nonsingular, and ΔB,A (M −1 ) = M −1 A−1 ΔA,B (M )M −1 A if the matrices A and M are nonsingular. Theorem 2.3. For any 5-tuple {A, B, C, M, N } of matrices of compatible sizes we have ∇A,C (M N ) = ∇A,B (M )N + M ∇B,C (N ), ΔA,C (M N ) = ΔA,B (M )N + AM ∇B,C (N ). Furthermore ΔA,C (M N ) = ΔA,B (M )N + AM BΔB −1 ,C (N ) if B is a nonsingular matrix and ΔA,C (M N ) = ΔA,B (M )N − AM ΔB,C −1 (N )C if C is a nonsingular matrix.

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363

Theorem 2.4. Represent the matrices A, B, M , ∇A,B (M ), and ΔA,B (M ) as 2 × 2 block matrices with blocks Ai,j , Bi,j , Mi,j , ∇i,j , and Δi,j , respectively, having compatible sizes (for i, j ∈ {0, 1}). Then ∇Aii ,Bjj (Mij ) = ∇ij − Ri,j , ΔAii ,Bjj (Mij ) = Δij + Si,j , where Ri,j = Mi,1−j B1−j,j − Ai,1−i M1−i,j , Si,j = Ai,i Mi,1−j B1−j,j + Ai,1−i M1−i,j Bj,j + Ai,1−i M1−i,1−j B1−j,j , for i, j ∈ {0, 1}. Remark 2.2. The expressions of Theorem 2.4 project the displacement generator of a matrix into those of its blocks so that the projection increases the length of the generator by at most rank(Ri,j ) or rank(Si,j ), that is, in both cases at most rank(Ai,1−i ) + rank(B1−j,j ). Hereafter (see Deﬁnitions 2.8–2.11) we only deal with diagonal and unit f -circulant operator matrices A and B; in both cases their blocks A1−i,i and B1−j,j have ranks zero or one (cf. Remark 2.3). In Section 3 for a nonsingular structured matrix M with dr(M ) = r we perform operations with short displacement generators to obtain a displacement generator of length r for its inverse. In the process of computing, the length of the generators can grow above the displacement rank, but then we compress the generators to the rank level based on the following results, valid in any ﬁeld. Theorem 2.5. Given a pair of n × l matrices G and H, it is suﬃcient to perform ˜ and H ˜ such that G ˜H ˜ T = GH T O(l2 n) ops to compute a pair of n × r matrices G T where r = rank(GH ) ≤ l. Proof. See [P01, Theorem 4.6.4]).

Corollary 2.1. Given a displacement generator of length l for a displacement operator L and an n × n matrix M with drL (M ) = r, it is suﬃcient to use O(l2 n) ops to compute a displacement generator of length r for the same pair of L and M . 2.5. Most popular matrices with displacement structure Toeplitz, Hankel, Vandermonde, and Cauchy matrices have displacement ranks one or two for appropriate operator matrices. These are most used matrices with displacement structure. Next we specify their natural extensions (see some other important classes in [P01, Examples 4.4.8 and 4.4.9]). Deﬁnition 2.8. T = (ti,j )ni,j=1 is a Toeplitz matrix if ti,j = ti+1,j+1 for every pair of its entries ti,j and ti+1,j+1 . Such matrix T is f -circulant for a scalar f if ti,j = f tk,l n wherever l − k + n = j − i > 0. In this case we write T = Zf (t) = h=1 th Zfh−1 where t = (th )nh=1 is the ﬁrst column of the matrix, th = th,1 , h = 1, . . . , n, and Zf is the unit f -circulant matrix with the ﬁrst column (0, 1, 0, . . . , 0)T and the ﬁrst row (0, . . . , 0, f ). Z0 (t) is the lower triangular Toeplitz matrix with the ﬁrst column t.

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is the reﬂection (or the unit Hankel) matrix if Deﬁnition 2.9. J = (jg,h )n−1,n−1 g,h=0 jg,n−1−g = 1 for g = 0, . . . , n − 1, jg,h = 0 for h + g = n − 1. (J (vi )n−1 i=0 = 2 , J = I.) H = (h ) is a Hankel matrix if h = h (vn−i−1 )n−1 i,j i,j i,j i−1,j+1 for i=0 every pair of its entries hi,j and hi−1,j+1 or equivalently if H = T J for a Toeplitz matrix T . Deﬁnition 2.10. For a positive integer n and a vector t = (ti )ni=1 , deﬁne the n × n )ni,j=1 . Vandermonde matrix V (t) = (tj−1 i Deﬁnition 2.11. For a positive integer n and two vectors s = (si )ni=1 and t = (ti )ni=1 such that the 2n scalars {si , tj }i,j are distinct, deﬁne the n × n Cauchy matrix 1 )i,j . C(s, t) = ( si −t j Fact 2.4. (See [P01, Chapters 2 and 3].) We have mS = O(m(n)) if S is an n × n Toeplitz or Hankel matrix for mS in Deﬁnition 2.6 and m(n) in Deﬁnition 2.1. mS = mS T = O(m(n) log n) if S is an n × n Vandermonde or Cauchy matrix. One can easily verify that Toeplitz, Hankel, Vandermonde, and Cauchy matrices have displacement ranks one or two for appropriate operator matrices. Fact 2.5. We have a) drZe ,ZfT (T ) ≤ 2 and drZe ,Zf (H) ≤ 2 for a Toeplitz matrix T , a Hankel matrix H, and a pair of scalars e and f , ef = 1, b) drD(t),Zf (V (t)) = 1 for a scalar f and a vector t = (ti )ni=1 such that tni f = 1, i = 1, . . . , n and c) drD(s),D(t) (C(s, t)) = 1 for a pair of vectors s and t with 2n distinct entries. We use the common nomenclatures of Toeplitz-like, Hankel-like, Vandermonde-like, and Cauchy-like matrices (or the matrices that have the structures of Toeplitz, Hankel, Vandermonde, and Cauchy types, respectively) to deﬁne the classes of matrices that have smaller displacement ranks under the same operators whose images have ranks one or two for Toeplitz, Hankel, Vandermonde, and Cauchy matrices, respectively. In view of our results in the previous section, this includes the transposes, blocks and inverses of the latter matrices, as well as the products, sums, and linear combinations of pairs of such matrices for the same or properly reconciled operators (see Theorems 2.2–2.4 and [P01, Chapter 4]). Next we equivalently deﬁne the matrices of these classes (in the memory eﬃcient way) as bilinear expressions via the entries of their displacements generators G = (gi )ri=1 and H = (hi )ri=1 (cf. [GO94] and [P01, Chapter 4]). Theorem 2.6. (Cf. [KKM79], [P01, Example 4.4.1].) A matrix T has displacement r generator ((gi )i , (hi )i ) under the operator ΔZe ,ZfT , that is T −Ze T ZfT = i=1 gi hTi (for 2r vectors gi and hi , i = 1, . . . , r, and for a pair of scalars e and f such that r ef = 1) if and only if T = i=1 Ze (gi )ZfT (hi ). Theorem 2.7. A matrix H has displacement ((gi )i , (hi )i ) under the r generator T operator ΔZe ,Zf , that is H − Ze HZf = g h (for 2r vectors gi and hi , i=1 i i i = 1,. . . , r, and for a pair of scalars e and f such that ef = 1) if and only if r H = i=1 Ze (gi )ZfT (Jhi )J.

Uniﬁed Nearly Optimal Algorithms Proof. Apply Theorem 2.6 for T = HJ and observe that JZfT J = Zf .

365

Theorem 2.8. (Cf. [P01, Example 4.4.6b].) A matrix V has displacement generator r ((gi )i , (hi )i ) under the operator ΔD(t),Zf , that is, V − diag(t)V Zf = i=1 gi hTi ) n n (for a vector t = (tj )j=1 and a scalar f such that ti f = 1 for i = 1, . . . , n) if and 1 n only if V = ri=1 diag( 1−t n )i=1 diag(gi )V (t)JZf (Jhi ). i

Theorem 2.9. (Cf. [P01, Example 4.4.1].) A matrix C has displacement generator ((gi )i , (hi )i ) under the operator ∇D(s),D(t) , that is diag(s)C − C diag(t) = r T n i=1 gi hi (for 2r vectors gi and hi , i = 1, . . . , r and two vectors s = (si )i=1 n and t= (ti )i=1 such that the 2n scalars {si , tj }i,j are distinct) if and only if r C = i=1 diag(gi )C(s, t) diag(hi ). Now let us assess our power in dealing with matrices having displacement structure. Theorems 2.2–2.4 enable us to express such matrices and the results of some operations with them in terms of their displacement generators (by using the order of rn parameters versus n2 entries). Theorems 2.6–2.9 enable us to go back from the displacement to matrices. The bilinear expressions in Theorems 2.6–2.9 reduce multiplication by a vector of a matrix with Toeplitz-like, Hankel-like, Vandermonde-like or Cauchy-like structures essentially to 2r multiplications of Toeplitz, Hankel, Vandermonde or Cauchy matrices by 2r vectors. Theorems 2.2–2.4 extend this property to transposes, inverses, sums, and products of such matrices as well as their blocks. In particular we specify some respective estimates for the arithmetic complexity in Corollary 2.2 below. Deﬁnition 2.12. Suppose we have a nonsingular n × n matrix M preprocessed with a procedure P that outputs ν parameters p1 , . . . , pν . Let iM (P ) be the minimum number of ops required to solve a linear system M x = f for any vector f provided a matrix M , its preprocessing P , and the parameters p1 , . . . , pν are ﬁxed. Write iM = minP {iM (P )} where the minimum is over all preprocessings P and write iM,h = minP (h) {iM (P )} where the minimum is over all preprocessings P = P (h) that amount to solving at most h linear systems of equations with the matrices M , M T and M T M . Corollary 2.2. We have mT = mH = O(lm(n)), mV = O(lm(n) log n), mC = O(lm(n) log n), iT,2l = iH,2l = O(lm(n)), iV,2l = O(lm(n) log n), and iC,2l = O(lm(n) log n) where T , H, V , and C stand for n × n matrices given with their displacement generators of lengths at most l and having structures of Toeplitz, Hankel, Vandermonde, and Cauchy types, respectively. The following result enables adjustment of the input displacement structure towards subsequent acceleration of the solution algorithms. In Section 3 we apply this method of displacement transformation (proposed in [P89/90] (cf. [P01, Sections 1.7, 4.8, and 4.9])) to accelerate the solution of linear systems with the structures of Vandermonde and Cauchy types by reducing them to linear systems with Toeplitz-like structures.

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Corollary 2.3. (Cf. [P89/90].) Given a positive integer r and a displacement generator of a length l for an n × n matrix M with the structure of Vandermonde (resp. Cauchy) type, one can generate matrices V1 and V2 that have structure of Vandermonde type and that are deﬁned with their displacement generators of lengths at most r and then apply O((l + r)m(n) log n) ops to compute a displacement generator of a length at most l + r (resp. l + 2r) for a Toeplitz-like matrix V1 M or M V2 (resp. V1 M V2 ). Proof. The corollary follows from Theorem 2.3 and Corollary 2.2.

Recall that in virtue of Theorem 2.1 the transition Δ ↔ ∇ between the operators Δ and ∇ cannot seriously aﬀect the displacement rank of a matrix. Likewise the simple estimates below imply that the modiﬁcation of the parameter f in the operator matrices Zf little aﬀects the displacement rank of a matrix. Fact 2.6. |drZe ,ZfT (T ) − drZe ,ZgT (T )| ≤ 1, |drZg ,ZfT (T ) − drZe ,ZfT (T )| ≤ 1,

(2.1)

|drZe ,Zf (H) − drZe ,Zg (H)| ≤ 1, |drZg ,Zf (H) − drZe ,Zf (H)| ≤ 1

(2.2)

for a 5-tuple (T, H, e, f, g). Remark 2.3. (Cf. Remark 2.2.) The rank of any oﬀ-diagonal block is zero for a diagonal matrix and is at most one for any matrix Zf . 2.6. Randomization Unlike deterministic algorithms, which always produce correct output, randomized algorithms produce correct output with a probability of at least 1 − for any ﬁxed positive tolerance . The randomized complexity estimates diﬀer depending whether they cover the cost of veriﬁcation of the correctness of computed solution (e.g., veriﬁcation that M x = f for the computed randomized solution x). If they cover this cost, they are of the Las Vegas type. (In this case at the estimated cost one either fails with a low probability or outputs the correct solution.) The other randomized complexity estimates are of the Monte Carlo type. (They cover algorithms whose output can be erroneous, but with a bounded low probability.) Given a nonsingular matrix in Z, what is the probability that it stays such in Zp for a random prime p in a ﬁxed large range, e.g., in (y/20, y] for a large integer y? Here is an estimate from [PMRW05], [PW08]. Theorem 2.10. Suppose that is a positive number, the matrix M ∈ Zn×n is nonsingular, and a prime p is randomly sampled from the range (y/20, y] under the uniform probability distribution in this range where y = nξ ln |M| ≥ 114, ξ = 16 ln 114 ln 114 16 ln 5.7−ln 114 = 16ν/(1 − ν) = 3.278885 . . ., and ν = 16 ln 5.7 = 0.17007650 . . .. Then P = Probability((det M ) mod p = 0) < .

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3. Computations in Zp with matrices having displacement structure 3.1. Inversion of strongly nonsingular structured matrices Theorem 3.1. Assume that a strongly nonsingular n × n matrix M in a ﬁeld F has structure of Toeplitz, Hankel, Vandermonde or Cauchy type (cf. Section 2.5), has a displacement rank r, and is given with its displacement generator of a length l. Then a displacement generator of the minimum length r for the matrix M −1 as well as the scalar det M can be computed by using O(l2 n + mM r log n) ﬁeld operations. Proof. The MBA divide-and-conquer algorithm by Morf 1974 and 1980 and Bitmead and Anderson 1980 [M74], [M80], and [BA80] was proposed for Toeplitz-like matrices. We adapt it to a more general class of matrices with displacement structure (cf. [OP98], [P01, Chapter 5]). Recall the well-known block triangular factorizations

−1 I 0 M00 0 M00 M01 M01 I M00 , (3.1) M= = −1 0 S M10 M11 I 0 I M10 M00

−1

−1 ˜ ˜ I 0 M M M00 M01 0 I −M00 . M −1 = ˜ 00 ˜ 01 = −1 0 S −1 0 I I −M10M00 M10 M11 (3.2) Here ˜ 10 = (−S −1 )(M10 M −1 ), M ˜ 01 = −(M −1 M01 )S −1 , M (3.3) 00 00 −1 −1 −1 ˜ ˜ ˜ 00 = M − (M M01 )M10 , M11 = S , (3.4) M 00

00

the k × k block matrix M00 and the (n − k) × (n − k) Schur complement −1 S = S(M, M00 ) = M11 − (M10 M00 )M01

(3.5)

are strongly nonsingular if so is the matrix M , and the sizes ni × ni of the block matrices Mii , i = 0, 1 are assumed to be balanced (say, n0 = %n/2&, n1 = n − n0 ). We obtain and recursively extend factorization (3.1) by applying the block Gauss– Jordan elimination to the matrix M and then recursively to the matrices M00 and S until the inversion problem is reduced to the case of one-by-one matrices. Actual computation goes back from the inverses of such one-by-one matrices to the matrix M −1 . In this recursive process we can also recursively factorize the scalar det M = (det M00 ) det S and then can compute it in n − 1 multiplications. To yield the claimed complexity bound, we maintain and exploit the structure of the input matrix M . In particular, we recursively compress the displacement generator of the matrix M and of all computed auxiliary matrices to the level of their displacement ranks and perform all computations with these matrices by operating with their displacement generators (cf. Theorems 2.2–2.5, Corollary 2.1, and Remark 2.2). Let us examine the associated operator matrices. Assume the pairs of operator matrices (A,B) for an input matrix M and (Aii ,Bjj ) for its submatrices Mij for i, j ∈ {0, 1}. Combine equations (3.3) and (3.4) with Theorems

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˜ ij 2.2–2.4 and obtain the pairs of operator matrices (Bii ,Ajj ) for the submatrices M −1 of M for i, j ∈ {0, 1} and hence obtain the pair of operator matrices (B,A) for the matrix M −1 such that drA,B (M ) = drB,A (M −1 ) (cf. Theorem 2.2). Likewise we arrive at consistent pairs of operator matrices for every matrix computed in the forward recursive process and for its inverse computed in the respective backward step. Now we can deduce the complexity bound claimed in Theorem 3.1 in the case of Sylvester displacements by combining Theorems 2.2 and 2.3 and our next result, in which we use Sylvester displacement ∇A,B (M ) (this allows singular operator matrices when we apply Theorem 2.2 for the inverses). Theorem 3.2. (Cf. Remark 3.1.) Assume Sylvester displacement ∇A,B (M ). Then a) all trailing principal blocks (that is Schur complements) computed in the forward recursive process of the adapted MBA algorithm have displacement ranks at most r + 4, b) all leading principal blocks processed in the forward recursive process of the adapted MBA algorithm have displacement ranks at most r +6, and c) all other matrices computed in the forward and backward recursive processes of the adapted MBA algorithm have displacement ranks at most 2r + 12. Proof. In the proof we will write S (k) (M ) to denote the (n − k) × (n − k) Schur complement S(M, M00 ) of the k × k block M00 in the n × n matrix M . We observe that S (g) (S (h) (M )) = S (g+h) (M ) because the MBA algorithm is a (structure preserving) variant of the block Gauss–Jordan elimination. Likewise S (h) (M (g+h) ) = (S (h) (M ))(g) . Therefore all trailing principal blocks computed in the MBA forward recursive process are Schur complements in the respective submatrices M (k) . Now part a) follows from Theorem 2.4 (applied in the case i = j) and Remark 2.2 because the inverse of every Schur complement is a trailing principal (that is, southeastern) block of the inverse of the matrix itself (cf. equation (3.4)) and because dr(N ) = dr(N −1 ) (cf. Theorem 2.2). Part b) follows from part a) and Theorem 2.4. Let us prove part c). In the ﬁrst step of the forward recursive MBA process we compute the oﬀ−1 −1 diagonal blocks M00 M01 and M10 M00 . In the next steps we compute similar −1 * −1 * *ij denote the (i, j)th blocks * * products M00 M01 and M01 M00 where the blocks M of the respective principal block computed in the previous step of the forward recursive process. As we have observed in the proof of part a), such a principal block is the matrix S (k) (M (h) )(g) for some integers g, h and k. Any of its blocks is also a block of the matrices M or S (k) (M (h) ) for some pair of h and k. Now combining part a) with Theorems 2.2–2.4 implies the bound dr(B) ≤ 2r + 12 claimed in −1 −1 part c). Indeed this bound surely covers the factors M10 M00 and M00 M01 of the −1 ˜ 01 of the inverse M , respectively (cf. equations (3.3)), but ˜ 10 and M blocks M the same bound is extended to the operands involved in the computation of the northwestern blocks computed in the backward process. Equations (3.4) and the −1 ˜ 00 ) ≤ dr(M −1 M01 ) + dr(M ˜ 00 )) support this extension inequalities dr((M00 M01 )M 00 at its ﬁnal step, and similar relationships support it at the other steps. The stronger upper bound r + 4 holds for the southeastern blocks computed in the backward

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process because they are the inverses of the Schur complements S (k) (M (h) ) for some integers h and k, and so we can apply Theorems 2.2 and 2.4. According to Theorem 3.2, displacement generators of all matrices involved −1 into the MBA process have lengths in O(r). For the ﬁnal transition from M00 and −1 −1 2 S to M (performed in terms of generators) we use A = O(rmM + r m(n)) = O(rmM ) ops. At the (i − 1)st preceding level of the recursion we perform similar operations with 2i matrices of sizes roughly (n/2i )×(n/2i ). For a positive constant c and mM ≥ cn this means O(rmM ) ops at each of the %log n& levels of the MBA backward recursion and thus O(rmM log n) ops overall. This completes the proof of Theorem 3.1 under Sylvester displacements. Theorem 2.1 enables extension to Stein displacements ΔA,B . We recall bounds (2.1) and (2.2) to treat the cases where A, B ∈ {Z0 , Z0T }. Remark 3.1. In the case of the Cauchy-like structure, the MBA recursive process involves only diagonal operator matrices, and so the bounds in Theorem 3.2 decrease to r in parts a) and b) and to 2r in part c). We have a smaller decrease in the case of the Vandermonde-like structure, where one half of the operator matrices in the MBA recursive process are diagonal. In the latter case and in the case of Toeplitz-like structure, we can choose the operator matrix B=Z0T , to decrease the bounds in Theorem 3.2 because the (1, 0)th block of this matrix is ﬁlled with zeros and thus has rank zero (cf. Remarks 2.2 and 2.3). Corollary 2.3 reduces the inversion of matrices with the structures of Vandermonde and Cauchy types to the inversion of Toeplitz-like matrices because M −1 = V2 (V1 M V2 )−1 V1 . This implies the following result. Corollary 3.1. The upper estimates of Theorem 3.1 can be decreased to O(l 2 n + m(n)r2 log n) ﬁeld operations for m(n) in Deﬁnition 2.1, even for matrices with the structures of Vandermonde and Cauchy types. 3.2. Inversion of nonsingular structured matrices in Zp Corollary 3.2. Assume a random prime p in the range (y/20, y] for a suﬃciently large integer y and a nonsingular matrix M ∈ Zn×n having structure of Toeplitz, Hankel, Vandermonde or Cauchy type, given with its displacement generator of a length l, and having a displacement rank r. Then a) the matrix M T M is expected to be strongly nonsingular in Zp , and b) if it is strongly nonsingular, then a displacement generator of length r for the matrix M −1 mod p can be computed by using O(l2 n + r2 m(n) log n) operations in Zp . Proof. The matrix M T M is strongly nonsingular in Z (cf. Deﬁnition 2.3) and is expected to stay such in Zp due to Theorem 2.10. This proves part a). Part b) follows from Corollary 3.1 for F = Zp and from the equation M −1 = (M T M )−1 M T .

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3.3. The case of singular matrices with displacement structure Let us extend our study to the case of singular input matrices M having a rank ρ and a displacement rank r. In this case we seek the inverse of a ρ × ρ nonsingular submatrix of the matrix M . Theorem 3.3. Assume that in a ﬁeld F an n × n matrix M of a rank ρ < n has generic rank proﬁle, has structure of (a) the Toeplitz or Hankel types or (b) Vandermonde or Cauchy types, has a displacement rank r, and is given with a displacement generator of a length l = O(r). Then (i) the rank ρ and (ii) a displacement generator of length r for the matrix (M (ρ) )−1 can be computed by using O(rmM (ρ) log ρ) ops in F, that is O(m(ρ)r2 log1+δ ρ) ops for δ = 0 in case (a) and δ = 1 in case (b). (iii) Within the same cost bound one can compute a solution x to a consistent linear system M x = f , in O(mM ) additional ops one can verify consistency of the system, and in O(rmM ) additional ops one can compute a shortest displacement generator for a matrix whose columns deﬁne a basis for the null space of the matrix M . Proof. Apply the adapted MBA algorithm as in the case of strongly nonsingular input matrices until it factorizes the submatrix M (ρ) and computes a shortest displacement generator (of length r + O(1)) for the matrix (M (ρ) )−1 . This takes O(rmM (ρ) log ρ) ops overall. Then the algorithm stops because it is prompted to invert the one-by-one matrix ﬁlled with the zero. To solve a consistent nonhomogeneous linear system M x = f , multiply the matrix (M (ρ) )−1 by the subvector made up of the ﬁrst ρ coordinates of the vector f and append n − ρ zero coordinates to the product to obtain a solution vector x. This stage involves O(mM (ρ) ) ops. To verify consistency of the nonhomogeneous linear system, multiply the matrix M by the vector x and compare the product with the vector f . In fact one only needs to multiply the (n − ρ) × ρ southwestern submatrix by the leading subvector of the dimension ρ in the vector x. If f = 0 and if we seek a solution x = (xi )ni=1 to the system M x = 0, then we substitute xn = 1 into this system and arrive at a nonhomogeneous linear system with n − 1 unknowns and equations. Finally (ρ) substitute into (3.1) and observe that the columns of the matrix (ρ) −1 M00 = M (M ) M01 form a basis for the null space of the matrix M . One can com−In−ρ pute a shortest displacement generator for the matrix (M (ρ) )−1 M01 in O(rmM ) additional ops (cf. Theorem 2.3). Theorem 3.4. I) To extend Theorem 3.3 to the case of input matrices not having generic rank proﬁle it is suﬃcient to perform O(mM ) additional ops, to generate 2n − 2 random parameters in the ﬁeld F, and to allow Monte Carlo randomization, that is, to allow erroneous output with a low probability. II) By performing additional O(rmM ) ops one yields Las Vegas randomization, that is, either fails with a low probability or arrives at the correct output. Proof. Part I) is implied by the following theorem.

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Theorem 3.5. Let a ﬁnite set S of a suﬃciently large cardinality |S| lie in a ﬁeld F and let a matrix M of a rank ρ ≤ n lie in Fn×n . Let {si , tj , i, j = 1, . . . , n} and {ui , vj , i, j = 1, . . . , n} be two sets of 2n distinct scalars each. Deﬁne randomized preprocessing of the matrix M overwriting it with the matrix XM Y (we j write M ← XM Y ) where X = Xg , Y = Yh for g, h ∈ {1, 2}, X1 = ( six−t )n , j i=1 yj n T Y1 = ( ui −vj )i=1 , X2 = Z0 (x), Y2 = Z0 (y), x1 = y1 = 1, and the other 2n − 2 coordinates of the vectors x = (xi )ni=1 and y = (yi )ni=1 are randomly sampled from the set S. Then both matrices X2 and Y2 are nonsingular and with a probability of at least (1 − n/|S|)2 both matrices X1 and Y1 are nonsingular. If the matrices X and Y are nonsingular, then with a probability of at least 1 − (ρ + 1)ρ/|S| matrix XM Y has generic rank proﬁle (and therefore is strongly nonsingular if the matrix M is nonsingular). Proof. See [KS91] on the case X = X2 , Y = Y2 and [P01, Corollary 5.6.3] on the case X = X1 , Y = Y1 . We can extend the displacement structure of the matrix M to the matrix XM Y by choosing appropriate matrices X = Xi and Y = Yi for i = 1, 2 to match the operator matrices associated with the matrix M . Then dr(XM Y ) ≤ dr(M )+2, a shortest displacement generator of the matrix XM Y is computed at a low cost (see Theorem 2.3), and so its recursive factorization and a shortest displacement generator for the matrix M −1 = Y (XM Y )−1 X are computed within the cost bounds of Corollary 3.1. This proves part I) of Theorem 3.4. To prove part II), verify correctness of the rank computation as follows: ﬁrst compute a displacement generator of length O(r) for the Schur complement S(XM Y, (XM Y )(ρ) ) of the nonsingular block (XM Y )(ρ) in the matrix XM Y (cf. equation (3.5) and Theorems 2.2, 2.3, and 3.3), then compress this generator to the minimum length (cf. Corollary 2.1), and ﬁnally verify that this length is zero. Similarly to Corollary 3.1, we reﬁne the estimates of Theorems 3.3 and 3.4 in the case (b). Corollary 3.3. In the upper estimates of Theorems 3.3 and 3.4 one can replace the bounds mM(ρ) ops with rm(ρ) and mM with rm(n), even in the case of input matrices with the structures of Vandermonde and Cauchy types, provided that the estimates are increased by rm(n) log n ops required for computing a shortest displacement generator for the matrix V1 M V2 where V1 and V2 are nonsingular Vandermonde or Vandermonde-like matrices (one of them is replaced with the identity matrix if the input matrix has structure of Vandermonde type). Clearly, the results of this subsection can be applied to matrices M in the ﬁeld F = Zn×n for any prime p. Furthermore for a large integer y and a random p prime p chosen in the range (y/20, y], a matrix M ∈ Zn×n is likely to keep its rank, displacement rank, and displacement generator in the transition from the ring Z to the ﬁeld Zp (cf. Theorem 2.10). Therefore the results of this subsection can be extended to structured integer matrices, as we specify next.

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4. Computations with structured integer matrices The algorithms in the previous section compute in Zp the determinant and a shortest generator for the inverse of a nonsingular n × n matrix M with displacement structure. We can repeat this computations for k distinct primes p1 , . . . , pk . The probability of failure and the overall number of ops increase at most by the factor of k. Then the Chinese Remainder Algorithm can produce the determinant and a shortest generator for the inverse modulo ki=1 pi . Suppose we let k = n and choose the primes p1 , . . . , pn at random in the range (y/20, y] for a value √ ξn ln(nα(M ))} (4.1) y > y0 = 20(2n)1/n n max{α(M ) n, for a ﬁxed positive and ξ < 4 in Theorem 2.10. Then we keep the probability of √ failure within and obtain det M and M −1 modulo p+ = ni=1 pi > 2n(α(M ) n)n . Fact 2.3 implies that p+ > 2| det M |, and so we can apply the recipe in Remark 2.1 to yield det M in Z and adj M in Zn×n . The overall computational cost is dominated at the stage of n applications of our algorithms of the previous section (see [GG03, Theorem 10.25] on the complexity of the Chinese Remainder Algorithm). In particular we arrive at the following result. Theorem 4.1. The asymptotic estimates of the previous section can be extended to computing the determinant and a shortest generator for the inverse of a strongly nonsingular n × n integer matrix M having displacement rank r and given with its displacement generator of a length l. The estimated numbers of random primes and ops increase by the factor n versus the estimates in the previous section. The ops are performed in the ﬁelds Zp1 , . . . , Zpn for n distinct primes p1 , . . . , pn chosen at random independently of each other in the range (y/20, y] where y satisﬁes (4.1). If l = O(r), then the overall cost bound turns into O(r2 nm(n) log n) ops performed with the precision of log y0 bits. This implies the overall Boolean cost bound of O((r2 nm(n) log n)μ(log y0 )) ops for y0 in (4.1) and m(n) and μ(d) in Deﬁnition 2.1. The ops are word operations if log2 y0 is within the word length. If log(1/) = O(log(nα(M ))), then the upper bound on the failure probability can be supported for log y0 = O(log(nα(M ))). If in addition log α(M ) = O(log n), then we can choose log y0 of the order of O(log n). The techniques in Sections 3.2 and 3.3 enable extension of all these estimates to computations with any nonsingular and singular integer matrices M having displacement structure at the additional randomized cost speciﬁed in Sections 3.2 and 3.3.

5. Related works Computations with matrices having displacement structure are closely linked to various fundamental polynomial computations [BGY80], [P01], [PMRa]. These links enable extension of our nearly optimal cost bounds from matrix to polynomial computations where the input values are integers. In particular links to

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Toeplitz computations enable extension of our cost bounds to the computation of the greatest common divisors, least common multiples, and Pad´e approximations for a pair of univariate polynomials and further to the Berlekamp–Massey problem of recovering the coeﬃcients of a linear recurrence from its values (see [PMRa]). On the extension to theoretical and practical acceleration of Wiedemann and block Wiedemann algorithms for determinants and Smith’s factors of a general matrix, see [P04a] and [PMRa]. For structured matrices M with log α(m) = O(log n) and r = O(1), our upper estimates on the Boolean cost are within the factor ˜ 2 log2 n) from the information lower bound of n2 log n. The upper bound has O(r been further decreased in [P02], [PMRa] by the factor of r log n based on combining Hensel’s symbolic lifting with numerical iterative reﬁnement and rational number reconstruction from their numerical approximation and from their values modulo a larger integers (cf. [PW02] and [WP03] on the latter subjects). Instead of k random primes for k of the order n, the lifting approach involves just a single prime. Moreover the paper [PMRa] elaborates upon lifting initialized with a power of two instead of customary basic primes p, so that the computations use the binary representation of all operands. The reader is referred to the paper [PMRa] on further details. Acknowledgement Criticism by Jesse Wolf helped us to detect and to ﬁx a number of defects in the original draft of this paper.

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[M74]

M. Morf, Fast Algorithms for Multivariable Systems, Ph.D. Thesis, Dept. Electrical Engineering, Stanford Univ., CA, 1974.

[M80]

M. Morf, Doubling Algorithms for Toeplitz and Related Equations, Proceedings of IEEE International Conference on ASSP, 954–959, IEEE Press, Piscataway, New Jersey, 1980.

[OP98]

V. Olshevsky, V.Y. Pan, A Uniﬁed Superfast Algorithm for Boundary Rational Tangential Interpolation Problem and for Inversion and Factorization of Dense Structured Matrices, Proc. 39th Ann. IEEE Symp. Foundation of Comp. Science (FOCS 98), 192–201, IEEE Computer Soc. Press, Los Alamitos, CA, 1998. [P84] V.Y. Pan, How Can We Speed up Matrix Multiplication?, SIAM Review, 26(3), 393–415, 1984. [P89/90] V.Y. Pan, On Computations with Dense Structured Matrices, Math. of Computation, 55(191), 179–190, 1990. Proceedings version in Proc. ISSAC’89, 34-42, ACM Press, New York, 1989. [P01] V.Y. Pan, Structured Matrices and Polynomials: Uniﬁed Superfast Algorithms, Birkh¨ auser/Springer, Boston/New York, 2001.

Uniﬁed Nearly Optimal Algorithms [P02]

375

V.Y. Pan, Can We Optimize Toeplitz/Hankel Computations? Proc. of the Fifth International Workshop on Computer Algebra in Scientiﬁc Computing (CASC’02), Yalta, Crimea, Sept. 2002 (E.W. Mayr, V.G. Ganzha, E.V. Vorozhtzov, Editors), 253–264, Technische Universit¨ at M¨ unchen, Germany, 2002. [P04a] V.Y. Pan, On Theoretical and Practical Acceleration of Randomized Computation of the Determinant of an Integer Matrix, Zapiski Nauchnykh Seminarov POMI (in English), 316, 163–187, St. Petersburg, Russia, 2004. Also available at http://comet.lehman.cuny.edu/vpan/ [PMRa] V.Y. Pan, B. Murphy, R.E. Rosholt, Uniﬁed Nearly Optimal Algorithms for Structured Integer Matrices and Polynomials, Technical Report 2008003, PhD Program in Computer Science, The Graduate Center, CUNY, New York, 2008. Available at http://www.cs.gc.cuny.edu/tr/techreport.php?id=352 V.Y. Pan, B. Murphy, R.E. Rosholt, Nearly Optimal Symbolic-Numerical Algorithms for Structured Integer Matrices and Polynomials, Proceedings of International Symposium on Symbolic-Numerical Computations (Kyoto, Japan, August 2009), (edited by Hiroshi Kai and Hiroshi Sekigawa), pp. 105–113, ACM Press, New York, 2009 [PMRW05] V.Y. Pan, B. Murphy, R.E. Rosholt, X. Wang, Toeplitz and Hankel Meet Hensel and Newton Modulo a Power of Two, Technical Report 2005008, PhD Program in Computer Science, The Graduate Center, CUNY, New York, 2005. Available at http://www.cs.gc.cuny.edu/tr/techreport.php?id=352 [PW02] V.Y. Pan, X. Wang, Acceleration of Euclidean Algorithm and Extensions, Proc. Intern. Symp. Symbolic and Algebraic Computation (ISSAC’02), 207– 213, ACM Press, New York, 2002. [PW08] V.Y. Pan, X. Wang, Degeneration of Integer Matrices Modulo an Integer, Linear Algebra and Its Application, 429, 2113–2130, 2008. [T64] W.F. Trench, An Algorithm for Inversion of Finite Toeplitz Matrices, J. of SIAM, 12, 515–4522, 1964. [T94] E.E. Tyrtyshnikov, How Bad Are Hankel Matrices? Numerische Mathematik, 67(2), 261–269, 1994. [WP03] X. Wang, V.Y. Pan, Acceleration of Euclidean Algorithm and Rational Number Reconstruction, SIAM J. on Computing, 32(2), 548–556, 2003. Victor Y. Pan Department of Mathematics and Computer Science Lehman College of the City University of New York Bronx, NY 10468, USA http://comet.lehman.cuny.edu/vpan/ e-mail: [email protected] Brian J. Murphy and Rhys Eric Rosholt Department of Mathematics and Computer Science Lehman College of the City University of New York Bronx, NY 10468, USA e-mail: [email protected] [email protected]

Operator Theory: Advances and Applications, Vol. 199, 377–396 c 2010 Birkh¨ auser Verlag Basel/Switzerland

V-cycle Optimal Convergence for DCT-III Matrices C. Tablino Possio Dedicated to Georg Heinig

Abstract. The paper analyzes a two-grid and a multigrid method for matrices belonging to the DCT-III algebra and generated by a polynomial symbol. The aim is to prove that the convergence rate of the considered multigrid method (V-cycle) is constant independent of the size of the given matrix. Numerical examples from diﬀerential and integral equations are considered to illustrate the claimed convergence properties. Mathematics Subject Classiﬁcation (2000). Primary 65F10, 65F15, 15A12. Keywords. DCT-III algebra, two-grid and multigrid iterations, multi-iterative methods.

1. Introduction In the last two decades, an intensive work has concerned the numerical solution of structured linear systems of large dimensions [6, 14, 16]. Many problems have been solved mainly by the use of (preconditioned) iterative solvers. However, in the multilevel setting, it has been proved that the most popular matrix algebra preconditioners are not eﬀective in general (see [23, 26, 20] and references therein). On the other hand, the multilevel structures often are the most interesting in practical applications. Therefore, quite recently, more attention has been focused (see [2, 1, 7, 5, 28, 9, 12, 10, 13, 22, 25, 19]) on the multigrid solution of multilevel structured (Toeplitz, circulants, Hartley, sine (τ class) and cosine algebras) linear systems in which the coeﬃcient matrix is banded in a multilevel sense and positive deﬁnite. The reason is due to the fact that these techniques are very eﬃcient, the total cost for reaching the solution within a preassigned accuracy being linear in the dimensions of the involved linear systems. The work of the author was partially supported by MIUR, grant number 2006017542.

378

C. Tablino Possio

In this paper we deal with the case of matrices generated by a polynomial symbol and belonging to the DCT-III algebra. This kind of matrices appears in the solution of diﬀerential equations and integral equations, see for instance [4, 18, 24]. In particular, they directly arise in certain image restoration problems or can be used as preconditioners for more complicated problems in the same ﬁeld of application [17, 18]. We consider the Two-Grid (TGM)/Multi-Grid (MGM) Method proposed and analyzed in [7] in terms of the algebraic multigrid theory developed by Ruge and St¨ uben [21] (for the foundation of the theory see [11] and [29]). Our aim is to provide general conditions under which the proposed MGM results to be optimally convergent with a convergence rate independent of the matrix size for a large class of matrices in the DCT-III algebra. We prove that for this class of matrices, the MGM results to be optimal in the sense of Deﬁnition 1.1 below, i.e., the problem of solving a linear system with coeﬃcient matrix A is asymptotically of the same cost as the direct problem of multiplying A by a vector. Deﬁnition 1.1. [3] Let {Am xm = bm } be a given sequence of linear systems with Am of size m. An iterative method for solving the systems Am xm = bm , m ∈ N+ , is optimal if 1. the arithmetic cost of each iteration is at most proportional to the complexity of a matrix vector product with matrix Am , 2. the number of iterations for reaching the solution within a ﬁxed accuracy can be bounded from above by a constant independent of m. A crucial role to prove our optimality result is played by the choice of the projection s operator Ps+1 which is used in the MGM to project from the level s to the level s + 1. In fact, the total cost of the proposed MGM will be of O(m) operations s since for any coarse level s we can ﬁnd a projection operator Ps+1 such that s costs O(ms ) operations where ms = • the matrix vector product involving Ps+1 m/2s ; s s • the coarse grid matrix Ams+1 = Ps+1 Ams (Ps+1 )T is also a matrix in the DCT-III algebra generated by a polynomial symbol and can be formed within O(ms ) operations; • the convergence rate of the MGM is independent of m. The paper is organized as follows. In Section 2 we brieﬂy recall TGM and MGM (standard V-cycle) and report the main tools regarding the convergence theory of algebraic multigrid methods [21]. In Section 3 we consider the TGM for matrices belonging to DCT-III algebra with reference to some optimal convergence properties, while Section 4 is devoted to the convergence analysis of its natural extension as V-cycle. Numerical evidences of the theoretical results are reported and discussed in Section 5, while Section 6 concerns complexity issues and conclusions.

V-cycle Optimal Convergence for DCT-III Matrices

379

2. Two-grid and multi-grid methods In this section we brieﬂy report the main results pertaining to the convergence theory of algebraic multigrid methods. Let us consider the generic linear system Am xm = bm with large dimension m, where Am ∈ Cm×m is a Hermitian positive deﬁnite matrix and xm , bm ∈ Cm . Let m0 = m > m1 > · · · > ms > · · · > msmin and let s Ps+1 ∈ Cms+1 ×ms

be a given full-rank matrix for any s. Lastly, let us denote by Vs a class of stationary iterative methods for linear systems of dimension ms . According to [11], the algebraic Two-Grid Method (TGM) is an iterative method whose generic step is deﬁned as follow. = T GM(s, xin xout s s , bs ) ν

pre xpre = Vs,pre (xin s s , bs )

Pre-smoothing iterations

rs = As xpre s − bs s rs+1 = Ps+1 rs s s As+1 = Ps+1 As (Ps+1 )H Solve As+1 ys+1 = rs+1 s H x ˆs = xpre s − (Ps+1 ) ys+1

Exact Coarse Grid Correction

ν

post xout = Vs,post (ˆ xs , bs ) s

Post-smoothing iterations

where we refer to the dimension ms by means of its subscript s. In the ﬁrst and last steps a pre-smoothing iteration and a post-smoothing iteration are applied νpre times and νpost times, respectively, according to the chosen stationary iterative method in the class Vs . Moreover, the intermediate steps deﬁne the so-called exact coarse grid cors rection operator, that depends on the considered projector operator Ps+1 . The global iteration matrix of the TGM is then given by T GMs CGCs

ν

post νpre = Vs,post CGCs Vs,pre ,

s s = Is − (Ps+1 )H A−1 s+1 Ps+1 As

(2.1) s s As+1 = Ps+1 As (Ps+1 )H ,

(2.2)

where Vs,pre and Vs,post denote the pre-smoothing and post-smoothing iteration matrices, respectively.

380

C. Tablino Possio

By means of a recursive procedure, the TGM gives rise to a Multi-Grid Method (MGM): the standard V-cycle is deﬁned as follows. = MGM(s, xin xout s s , bs ) if s ≤ smin

then Solve As xout = bs s

Exact solution

else ν

pre in xpre s = Vs,pre (xs , bs )

Pre-smoothing iterations

rs = As xpre ss − bs rs+1 = Ps+1 rs ys+1 = MGM(s + 1, 0s+1 , rs+1 ) s H x ˆs = xpre s − (Ps+1 ) ys+1

Coarse Grid Correction

ν

post xout = Vs,post (ˆ xs , bs ) s

Post-smoothing iterations

s s As (Ps+1 )H are more proﬁtably Notice that in MGM the matrices As+1 = Ps+1 formed in the so-called setup phase in order to reduce the computational costs. The global iteration matrix of the MGM can be recursively deﬁned as

M GMsmin

=

O ∈ Csmin ×smin ,

M GMs

=

νpre νpost s s Is − (Ps+1 Vs,post )H (Is+1 − M GMs+1 ) A−1 s+1 Ps+1 As Vs,pre , s = smin − 1, . . . , 0.

Hereafter, by · 2 we denote the Euclidean norm on Cm and the associated induced matrix norm over Cm×m . If X is positive deﬁnite, · X = X 1/2 · 2 denotes the Euclidean norm weighted by X on Cm and the associated induced matrix norm. Finally, if X and Y are Hermitian matrices, then the notation X ≤ Y means that Y − X is nonnegative deﬁnite. Some general conditions that ensure the convergence of an algebraic TGM and MGM are due to Ruge and St¨ uben [21]. Theorem 2.1 (TGM convergence [21]). Let m0 , m1 be integers such that m0 > m1 > 0, let A ∈ Cm0 ×m0 be a positive deﬁnite matrix. Let V0 be a class of iterative methods for linear systems of dimension m0 and let P10 ∈ Cm1 ×m0 be a given fullrank matrix. Suppose that there exist αpost > 0 independent of m0 such that V0,post x2A ≤ x2A − αpost x2AD−1 A

for any x ∈ Cm0

(2.3)

V-cycle Optimal Convergence for DCT-III Matrices

381

(where D is the diagonal matrix formed by the diagonal entries of A) and that there exists γ > 0 independent of m0 such that min x − (P10 )H y2D ≤ γx2A

for any x ∈ Cm0 .

y∈Cm1

Then, γ ≥ αpost and T GM0 A ≤

7 1 − αpost /γ.

(2.4)

(2.5)

It is worth stressing that in Theorem 2.1 the matrix D ∈ Cm0 ×m0 can be substituted by any Hermitian positive deﬁnite matrix X: clearly the choice X = I can give rise to valuable simpliﬁcations [2]. At ﬁrst sight, the MGM convergence requirements are more severe since the smoothing and CGC iteration matrices are linked in the same inequalities as stated below. Theorem 2.2 (MGM convergence [21]). Let m0 = m > m1 > m2 > · · · > ms > s · · · > msmin and let A ∈ Cm×m be a positive deﬁnite matrix. Let Ps+1 ∈ Cms+1 ×ms be full-rank matrices for any level s. Suppose that there exist δpre > 0 and δpost > 0 such that 2

2

2

νpre νpre Vs,pre xA ≤ xAs − δpre CGCs Vs,pre xA s

2

2

2

νpost xA ≤ xAs − δpost CGCs xAs Vs,posr s

s

for any x ∈ Cms

(2.6a)

for any x ∈ Cms

(2.6b)

both for each s = 0, . . . , smin − 1, then δpost ≤ 1 and F 1 − δpost M GM0 A < 1. 1 + δpre (k)

(2.7)

By virtue of Theorem 2.2, the sequence {xm }k∈N will converge to the solution of the linear system Am xm = bm and within a constant error reduction not depending on m and smin if at least one between δpre and δpost is independent of m and smin . Nevertheless, as also suggested in [21], the inequalities (2.6a) and (2.6b) can be split as ⎧ ν 2 νpre 2 pre ⎪ ⎨Vs,pre xAs ≤ xAs − α Vs,pre xAs Ds−1 As 2 2 (2.8) CGCs xAs ≤ γ xAs Ds−1 As ⎪ ⎩ δpre = α/γ and ⎧ νpost 2 2 2 ⎪ ⎨Vs,post xAs ≤ xAs − β xAs Ds−1 As 2 2 (2.9) CGCs xAs ≤ γ xAs Ds−1 As ⎪ ⎩ δpost = β/γ where Ds is the diagonal matrix formed by the diagonal entries of As (again, the AD −1 A-norm is not compulsory [2] and the A2 -norm will be considered in the following) and where, more importantly, the coeﬃcients α, β and γ can diﬀer in each recursion level s since the step from (2.8) to (2.6a) and from (2.9) to (2.6b) are purely algebraic and does not aﬀect the proof of Theorem 2.2.

382

C. Tablino Possio

Therefore, in order to prove the V-cycle optimal convergence, it is possible to consider the inequalities 2

2

2

νpre νpre Vs,pre xA ≤ xAs − αs Vs,pre xA2 s

ν

2

s

2

2

post xA ≤ xAs − βs xA2s Vs,post s

2

2

CGCs xAs ≤ γs xA2s

for any x ∈ Cms

(2.10a)

for any x ∈ Cms

(2.10b)

for any x ∈ Cms .

(2.10c)

where it is required that αs , βs , γs ≥ 0 for each s = 0, . . . , smin − 1 and δpre =

min

0≤s<smin

αs , γs

δpost =

min

0≤s<smin

βs . γs

(2.11)

We refer to (2.10a) as the pre-smoothing property, (2.10b) as the post-smoothing property and (2.10c) as the approximation property (see [21]). An evident beneﬁt in considering the inequalities (2.10a)–(2.10c) relies on the fact that the analysis of the smoothing iterations is distinguished from the more diﬃcult analysis of the projector operator. Moreover, the MGM smoothing properties (2.10a) and (2.10b) represent a generalization of the TGM smoothing property (2.3) with D substituted by I, in accordance with the previous reasoning.

3. Two-grid and multi-grid methods for DCT-III matrices Let Cm = {Cm ∈ Rm×m |Cm = Qm Dm QTm } the one-level DCT-III cosine matrix algebra, i.e., the algebra of matrices that are simultaneously diagonalized by the orthogonal transform %@ 2 8&m 2 − δj,1 (i − 1)(j − 1/2)π (3.1) Qm = cos m m i,j=1

with δi,j denoting the Kronecker symbol. Let f be a real-valued even trigonometric polynomial of degree k and period 2π. Then, the DCT-III matrix of order m generated by f is deﬁned as (j − 1)π [m] [m] . Cm (f ) = Qm Dm (f )QTm , Dm (f ) = diag1≤j≤m f xj , xj = m Clearly, Cm (f ) is a symmetric band matrix of bandwidth 2k + 1. In the following, we denote in short with Cs = Cms (gs ) the DCT-III matrix of size ms generated by the function gs . An algebraic TGM/MGM method for (multilevel) DCT-III matrices generated by a real-valued even trigonometric polynomial has been proposed in [7]. Here, we brieﬂy report the relevant results with respect to TGM convergence analysis, the aim being to prove in Section 4 the V-cycle optimal convergence under suitable conditions. s Indeed, the projector operator Ps+1 is chosen as s s Ps+1 = Ts+1 Cs (ps )

V-cycle Optimal Convergence for DCT-III Matrices

383

s ∈ Rms+1×ms , ms+1 = ms /2, is the cutting operator deﬁned as where Ts+1 2 √ s for j ∈ {2i − 1, 2i}, i = 1, . . . , ms+1 , 1/ 2 Ts+1 i,j = 0 otherwise,

(3.2)

and Cs (ps ) is the DCT-III cosine matrix of size ms generated√by a suitable even trigonometric polynomial ps . Here, the scaling by a factor 1/ 2 is introduced in s order to normalize the matrix Ts+1 with respect to the Euclidean norm. From the point of view of an algebraic multigrid this is a natural choice, while in a geometric multigrid it is more natural to consider just a scaling by 1/2 in the projector, to obtain an average value. The cutting operator plays a leading role in preserving both the structural and spectral properties of the projected matrix Cs+1 : in fact, it ensures a spectral link between the space of the frequencies of size ms and the corresponding space of frequencies of size ms+1 , according to the following Lemma. s Lemma 3.1. [7] Let Qs ∈ Rms ×ms and Ts+1 ∈ Rms+1 ×ms be given as in (3.1) and (3.2), respectively. Then s Ts+1 Qs = Qs+1 [Φs+1 , Θs+1 Πs+1 ],

where Φs+1

Θs+1

(3.3)

%

[ms ] & 1 xj (j − 1)π [m ] , xj s = = diagj=1,...,ms+1 cos , 2 2 ms % [ms ] & 1 xj π + , = diagj=1,...,ms+1 − cos 2 2 2

(3.4a)

(3.4b)

and Πs+1 ∈ Rms+1×ms+1 is the permutation matrix given by ⎤ ⎡ 1 0 ... 0 ⎢ 0 0 1 ⎥ ⎥ ⎢ ⎥. ⎢ .. . ⎦ ⎣ . .. 0 1 0 As a consequence, let As = Cs (fs ) be the DCT-III matrix generated by fs , then s s As+1 = Ps+1 As (Ps+1 )T = Cs+1 (fs+1 )

where fs+1 (x)

=

x/2 x 2 x cos fs p 2 2 s 2

x 2 x π − x/2 fs π − ps π − , + cos2 2 2 2 2

(3.5) x ∈ [0, π].

384

C. Tablino Possio

On the other side, the convergence of the proposed TGM at size ms is ensured by choosing the polynomial as follows. Let x0 ∈ [0, π) be a zero of the generating function fs . The polynomial ps is chosen so that lim0

x→x

p2s (π − x) < +∞, fs (x)

p2s (x) + p2s (π − x) > 0.

(3.6a) (3.6b)

In the special case x0 = π, the requirement (3.6a) is replaced by lim

x→x0 =π

p2s (π − x) < +∞. cos2 x2 fs (x)

(3.7a)

If fs has more than one zero in [0, π], then ps will be the product of the polynomials satisfying the condition (3.6a) (or (3.7a)) for every single zero and globally the condition (3.6b). It is evident that the polynomial ps must have zeros of proper order in any mirror point x ˆ0 = π − x0 , where x0 is a zero of fs . It is worth stressing that conditions (3.6a) and (3.6b) are in perfect agreement with the case of other structures such as τ , symmetric Toeplitz and circulant matrices (see, e.g., [22, 25]), while the condition (3.7a) is proper of the DCT-III algebra and it corresponds to a worsening of the convergence requirements. Moreover, as just suggested in [7], in the case x0 = 0 the condition (3.6a) can also be weakened as 2 ps (π − x) cos2 π−x 2 < +∞. (3.8a) lim fs (x) x→x0 =0 We note that if fs is a trigonometric polynomial of degree k, then fs can have a zero of order at most 2k. If fs (π) = 0, then by (3.6a) the degree of ps has to be less than or equal to %k/2&. If fs (π) = 0, then the degree of ps is less than or equal to %(k + 1)/2&. Notice also that from (3.5) it is easy to obtain the Fourier coeﬃcients of fs+1 and hence the nonzero entries of As+1 = Cs+1 (fs+1 ). In addition, we can obtain the zeros of fs+1 and their orders by knowing the zeros of fs and their orders. Lemma 3.2. [7] If 0 ≤ x0 ≤ π/2 is a zero of fs , then by (3.6a), ps (π − x0 ) = 0 and hence by (3.5), fs+1 (2x0 ) = 0, i.e., y 0 = 2x0 is a zero of fs+1 . Furthermore, because ps (π − x0 ) = 0, by (3.6b), ps (x0 ) > 0 and hence the orders of x0 and y 0 are the same. Similarly, if π/2 ≤ x0 < π, then y 0 = 2(π − x0 ) is a zero of fs+1 with the same order as x0 . Finally, if x0 = π, then y 0 = 0 with order equal to the order of x0 plus two. In [7] the Richardson method has be considered as the most natural choice for the smoothing iteration, since the corresponding iteration matrix Vm := Im − ωAm ∈ Cm×m belongs to the DCT-III algebra, too. Further remarks about such a type of smoothing iterations and the tuning of the parameter ω are reported in [25, 1].

V-cycle Optimal Convergence for DCT-III Matrices

385

Theorem 3.3. [7] Let Am0 = Cm0 (f0 ) with f0 being a nonnegative trigonometric polynomial and let Vm0 = Im0 − ωAm0 with ω = 1/f0∞ for the post-smoothing iteration. Then, under the quoted assumptions and deﬁnitions the inequalities (2.3) and (2.4) hold true, and the proposed TGM converges linearly. Here, it could be interesting to come back to some key steps in the proof of the quoted Theorem 3.3 in order to highlight the structure with respect to any point and its mirror point according to the considered notations. By referring to a proof technique developed in [22], the claimed thesis is obtained by proving that the right-hand sides in the inequalities

1 π − x p2s (π − x) γ≥ cos2 , (3.9a) ds (x) 2 fs (x)

x p2 (x) π − x p2s (π − x) 1 s cos2 + cos2 , (3.9b) γ≥ ds (x) 2 fs (x) 2 fs (π − x)

x π−x ds (x) = cos2 (3.9c) p2s (x) + cos2 p2s (π − x) 2 2 are uniformly bounded on the whole domain so that γ is a universal constant. It is evident that (3.9a) is implied by (3.9b). Moreover, both the two terms in (3.9b) and in ds (x) can be exchanged each other, up to the change of variable y = π − x. Therefore, if x0 = π it is evident that (3.6a) and (3.6b) ensure the required uniform boundedness since the condition p2s (x) + p2s (π − x) > 0 implies ds (x) > 0. In the case x0 = π, the inequality (3.9b) can be rewritten as % & p2s (π − x) 1 p2s (x) γ≥ + (3.10) p2s (x) p2 (π − x) cos2 x2 fs (x) cos2 π−x fs (π − x) 2 π−x + s x cos2 2 cos2 2 so motivating the special case reported in (3.7a).

4. V-cycle optimal convergence In this section we propose a suitable modiﬁcation of (3.6a), or (3.7a), with respect to the choice of the polynomial involved into the projector, that allows us to prove the V-cycle optimal convergence according to the veriﬁcation of the inequalities (2.10a)–(2.10c) and the requirement (2.11). Thanks to the choice D = I, it is worth stressing that the MGM smoothing properties do not require a true veriﬁcation, since the proofs of (2.10a) and (2.10b) are identical in any matrix algebra with unitary transforms. Proposition 4.1. Let As = Cms (fs ) for any s = 0, . . . , smin , with fs ≥ 0, and let ωs be B we choose αs and βs such that αs ≤ A such that 0 < ωs ≤ 2/fs ∞ . If ωs min 2, (2 − ωs fs ∞ )/(1 − ωs fs ∞ )2 and βs ωs (2 − ωs fs ∞ ) then for

386

C. Tablino Possio

any x ∈ Cm the inequalities Vs,pre x2As

≤

x2As − αs Vs,pre x2A2s

(4.1)

Vs,post x2As

≤

x2As − βs x2A2s

(4.2)

hold true. Notice, for instance, that the best bound to βs is given by 1/fs ∞ and it is obtained by taking ωs = 1/fs∞ [25, 1]. Concerning the analysis of the approximation condition (2.10c) we consider here the case of a generating function f0 with a single zero at x0 . In such a case, the choice of the polynomial in the projector is more severe with respect to the case of TGM. Let x0 ∈ [0, π) a zero of the generating function fs . The polynomial ps is chosen in such a way that lim0

x→x

ps (π − x) < +∞, fs (x)

p2s (x) + p2s (π − x) > 0.

(4.3a) (4.3b)

In the special case x0 = π, the requirement (4.3a) is replaced by lim 0

x→x =π

ps (π − x) < +∞. cos x2 fs (x)

(4.4a)

Notice also that in the special case x0 = 0 the requirement (4.3a) can be weakened as cos π−x ps (π − x) 2 lim < +∞. (4.5a) 0 x→x =0 fs (x) Proposition 4.2. Let As = Cms (fs ) for any s = 0, . . . , smin , with fs ≥ 0. Let s s Ps+1 = Ts+1 Cs (ps ), where ps (x) is fulﬁlling (4.3a) (or (4.4a)) and (4.3b). Then, for any s = 0, . . . , smin − 1, there exists γs > 0 independent of ms such that CGCs x2As ≤ γs x2A2s

for any x ∈ Cms ,

(4.6)

where CGCs is deﬁned as in (2.2). Proof. Since s s s s CGCs = Is − (Ps+1 )T (Ps+1 As (Ps+1 )T )−1 Ps+1 As

is a unitary projector, it holds that CGCsT As CGCs = As CGCs . Therefore, the target inequality (4.6) can be simpliﬁed and symmetrized, giving rise to the equivalent matrix inequality s T s s T −1 s s = Is − A1/2 Ps+1 A1/2 ≤ γs As . CGC s (Ps+1 ) (Ps+1 As (Ps+1 ) ) s

(4.7)

s Qs can be permuted into a 2 × 2 block Hence, by invoking Lemma 3.1, QTs CGC diagonal matrix whose jth block, j = 1, . . . , ms+1 , is given by the rank-1 matrix

V-cycle Optimal Convergence for DCT-III Matrices (see [8] for the analogous τ case) 1 I2 − 2 cj + s2j where cj = cos

[m ]

xj s 2

c2j cj sj

cj sj s2j

2

p

[m ] f (xj s )

sj = − cos

387

,

[ms ]

π − xj 2

[ms ]

p2 f (π − xj

).

As in the proof of the TGM convergence, due to the continuity of fs and ps , (4.7) is proven if the right-hand sides in the inequalities

π − x p2s fs (π − x) 1 cos2 (4.8a) γs ≥ 2 fs (x) d s (x)

2 p2 f (x) ps fs (π − x) 1 s s 2 π−x 2 x cos + cos (4.8b) γs ≥ 2 f (x) 2 f (π − x) s s ds (x)

x π−x d s (x) = cos2 (4.8c) p2s fs (x) + cos2 p2s f (π − x) 2 2 are uniformly bounded on the whole domain so that γs are universal constants. Once again, it is evident that (4.8a) is implied by (4.8b). Moreover, both the terms in (4.8b) and in d s (x) can be exchanged each other, up to the change of variable y = π − x. Therefore, if x0 = π, (4.8b) can be rewritten as

p2 (x) 1 π − x p2s (π − x) s 2 x cos2 γs ≥ + cos (4.9) 2 fs2 (x) 2 fs2 (π − x) dˆs (x) where

x p2 (x) π − x p2s (π − x) s + cos2 , dˆs (x) = cos2 2 fs (π − x) 2 fs (x) so that (4.3a) and (4.3b) ensure the required uniform boundedness. In the case x0 = π, the inequality (4.8b) can be rewritten as % p2s (π − x) 1 γs ≥ 2 2 p (x) ps (π − x) cos2 x2 fs2 (x) π−xs + cos2 2 fs (π − x) cos2 x2 fs (x) p2 (x) + 2 π−xs 2 fs (π − x) cos 2 so motivating the special case reported in (4.4a).

& (4.10)

Remark 4.3. Notice that in the case of pre-smoothing iterations and under the assumption Vs,pre nonsingular, the approximation condition νpre νpre x2A ≤ γs Vs,pre x2A2 for any x ∈ Cms , CGCs Vs,pre s

s

(4.11)

388

C. Tablino Possio

s ≤ γs As obtained in Propois equivalent to the condition, in matrix form, CGC sition 4.2. In Propositions 4.1 and 4.2 we have obtained that for every s (independent of m = m0 ) the constants αs , βs , and γs are absolute values not depending on m = m0 , but only depending on the functions fs and ps . Nevertheless, in order to prove the MGM optimal convergence according to Theorem 2.2, we should verify at least one between the following inf–min conditions [2]: δpre = inf

min

m0 0≤s≤log2 (m0 )

αs > 0, γs

δpost = inf

min

m0 0≤s≤log2 (m0 )

βs > 0. γs

(4.12)

First, we consider the inf-min requirement (4.12) by analyzing the case of a generating function f˜0 with a single zero at x0 = 0. It is worth stressing that in such a case the DCT-III matrix A˜m0 = Cm0 (f˜0 ) is [m ] singular since 0 belongs to the set of grid points xj 0 = (j−1)π/m0 , j = 1, . . . , m0 . Thus, the matrix A˜m0 is replaced by eeT [m ] Am0 = Cm0 (f0 ) = Cm0 (f˜0 ) + f˜0 x2 0 · m0 with e = [1, . . . , 1]T ∈ Rm0 and where the rank-1 additional term is known as Strang correction [30]. Equivalently, f˜0 ≥ 0 is replaced by the generating function [m ] f0 = f˜0 + f˜0 x2 0 χw[ms ] +2πZ > 0, (4.13) 1

[m ]

where χX is the characteristic function of the set X and w1 0 = x0 = 0. In Lemma 4.4 below, it is reported the law to which the generating functions are subjected at the coarser levels. With respect to this target, it is useful to consider the following factorization result: let f ≥ 0 be a trigonometric polynomial with a single zero at x0 of order 2q. Then, there exists a positive trigonometric polynomial ψ such that q

f (x) = [1 − cos(x − x0 )] ψ(x).

(4.14)

Notice also that, according to Lemma 3.2, the location of the zero is never shifted at the subsequent levels. Lemma 4.4. Let f0 (x) = f˜0 (x) + c0 χ2πZ (x), with f˜0 (x) = [1 − cos(x)]q ψ0 (x), q being a positive integer and ψ0 being a positive trigonometric polynomial and with [m0 ] ˜ . Let ps (x) = [1 + cos(x)]q for any s = 0, . . . , smin − 1. Then, under c0 = f0 x2 the same assumptions of Lemma 3.1, each generating function fs is given by fs (x) = f˜s (x) + cs χ2πZ (x),

f˜s (x) = [1 − cos(x)]q ψs (x).

The sequences {ψs } and {cs } are deﬁned as ψs+1 = Φq,ps (ψs ),

cs+1 = cs p2s (0),

s = 0, . . . , smin − 1,

V-cycle Optimal Convergence for DCT-III Matrices

389

where Φq,p is an operator such that x 1 ' x ( [Φq,p (ψ)] (x) = q+1 (ϕ p ψ) + (ϕ p ψ) π − , (4.15) 2 2 2 with ϕ(x) = 1 + cos(x). Moreover, each f˜s is a trigonometric polynomial that vanishes only at 2πZ with the same order 2q as f˜0 . Proof. The claim is a direct consequence of Lemma 3.1. Moreover, since the func tion ψ0 is positive by assumption, the same holds true for each function ψs . Hereafter, we make use of the following notations: for a given function f , we will write Mf = supx |f |, mf = inf x |f | and μ∞ (f ) = Mf /mf . Now, if x ∈ (0, 2π) we can give an upper bound for the left-hand side R(x) in (4.9), since it holds that 2 cos2 x2 p2s (x) cos2 π−x ps (π − x) 2 + 2 2 fs (π − x) f (x) s 2 R(x) = cos2 x2 p2s (x) cos2 π−x ps (π − x) 2 + fs (π − x) fs (x) cos2 π−x cos2 x2 2 + ψs2 (π − x) ψs2 (x) = 2 π−x 2 π−x cos ps (x) cos ps (π − x) 2 2 + ψs (π − x) ψs (x) 1 M ψs Mψs x π−x , ≤ ≤ 2 2 2 mψs cos 2 ps (x) + cos m2ψs ps (π − x) 2 we can consider γs = Mψs /m2ψs . In the case x = 0, since ps (0) = 0, it holds R(0) = 1/fs(π), so that we have also to require 1/fs (π) ≤ γs . However, since 1/fs (π) ≤ Mψs /m2ψs , we take γs∗ = Mψs /m2ψs as the best value. ∗ −1 q In (2.9), by choosing ωs∗ = fs −1 ∞ , we simply ﬁnd βs = fs ∞ ≥ 1/(2 Mψs ) and as a consequence, we obtain m2ψs 1 1 βs∗ ≥ q · = q 2 . ∗ γs 2 Mψs Mψs 2 μ∞ (ψs )

(4.16)

A similar relation can be found in the case of a pre-smoothing iteration. Nevertheless, since it is enough to prove one between the inf-min conditions, we focus our attention on condition (4.16). So, to enforce the inf-min condition (4.12), it is enough to prove the existence of an absolute constant L 5 such that μ∞ (ψs ) L < +∞ uniformly in order to deduce that M GM0 A0 1 − (2q L2 )−1 < 1. Proposition 4.5. Under the same assumptions of Lemma 4.4, let us deﬁne ψs = [Φps ,q ]s (ψ) for every s ∈ N, where Φp,q is the linear operator deﬁned as in (4.15). Then, there exists a positive polynomial ψ∞ of degree q such that ψs uniformly converges to ψ∞ , and moreover there exists a positive real number L such that μ∞ (ψs ) L for any s ∈ N.

390

C. Tablino Possio

Proof. Due to the periodicity and to the cosine expansions of all the involved functions, the operator Φq,p in (4.15) can be rewritten as x 1 ' x ( [Φq,p (ψ)] (x) = q+1 (ϕ p ψ) + (ϕ p ψ) π + . (4.17) 2 2 2 The representation of Φq,p in the Fourier basis (see Proposition 4.8 in [2]) leads to an operator from Rm(q) to Rm(q) , m(q) proper constant depending only on q, ¯ q in equation (4.14) of which is identical to the irreducible nonnegative matrix Φ [2], with q + 1 in place of q. As a consequence, the claimed thesis follows by referring to the Perron– Frobenius theorem [15, 31] according to the very same proof technique considered in [2]. Lastly, by taking into account all the previous results, we can claim the optimality of the proposed MGM. Theorem 4.6. Let f˜0 be an even nonnegative trigonometric polynomial vanishing at 0 with order 2q. Let m0 = m > m1 > · · · > ms > · · · > msmin , ms+1 = ms /2. For s any s = 0, . . . , smin − 1, let Ps+1 be as in Proposition 4.2 with ps (x) = [1 + cos(x)]q , and let Vs,post = Ims − Ams /fs∞ . If we set Am0 = Cm0 (f˜0 + c0 χ2πZ ) with c0 = [m ] f˜0 (w2 0 ) and we consider b ∈ Cm0 , then the MGM (standard V-cycle) converges to the solution of Am0 x = b and is optimal (in the sense of Deﬁnition 1.1). Proof. Under the quoted assumptions it holds that f˜0 (x) = [1 − cos(x)]q ψ0 (x) for some positive polynomial ψ0 (x). Therefore, it is enough to observe that the optimal convergence of MGM as stated in Theorem 2.2 is implied by the inf-min condition (4.12). Thanks to (4.16), the latter is guaranteed if the quantities μ∞ (ψs ) are uniformly bounded and this holds true according to Proposition 4.5. Now, we consider the case of a generating function f0 with a unique zero at x0 = π, this being particularly important in applications since the discretization of certain integral equations leads to matrices belonging to this class. For instance, the signal restoration leads to the case of f0 (π) = 0, while for the super-resolution problem and image restoration f0 (π, π) = 0 is found [5]. By virtue of Lemma 3.2 we simply have that the generating function f1 related to the ﬁrst projected matrix uniquely vanishes at 0, i.e., at the ﬁrst level the MGM projects a discretized integral problem, into another which is spectrally and structurally equivalent to a discretized diﬀerential problem. With respect to the optimal convergence, we have that Theorem 2.2 holds ¯ since δ results to be a constant and independent of true with δ = min{δ0 , δ} m0 . More precisely, δ0 is directly related to the ﬁnest level and δ¯ is given by the inf-min condition of the diﬀerential problem obtained at the coarser levels. The latter constant value has been previously shown, while the former can be proven as follows. We are dealing with f0 (x) = (1 + cos(x))q ψ0 (x) and according to (4.3a) we

V-cycle Optimal Convergence for DCT-III Matrices

391

[m ] choose p˜0 (x) = p0 (x) + d0 χ2πZ with p0 (x) = (1 + cos(x))q+1 and d0 = p0 (w2 0 ). ˜ Therefore, an upper bound for the left-hand side R(x) in (4.10) is obtained as

Mψ ˜ R(x) ≤ 20, mψ0 i.e., we can consider γ0 = Mψ0 /m2ψ0 and so that a value δ0 independent of m0 is found.

5. Numerical experiments Hereafter, we give numerical evidence of the convergence properties claimed in the previous sections, both in the case of proposed TGM and MGM (standard V-cycle), for two types of DCT-III systems with generating functions having zero at 0 (diﬀerential-like problems) and at π (integral-like problems). s The projectors Ps+1 are chosen as described in Section 3 and in Section 4. The Richardson smoothing iterations are used twice in each iteration with ω = 2/f ∞ and ω = 1/f ∞, respectively. The iteration is stopped when the Euclidean norm of the relative residual at dimension m0 is less than 10−7. Moreover, the exact solution of the system is found by a direct solver when the coarse grid dimension equals to 16 (162 in the additional two-level tests). 5.1. Case x0 = 0 (diﬀerential-like problems) First, we consider the case Am = Cm (f0 ) with f0 (x) = [2 − 2 cos(x)]q , i.e., with a unique zero at x0 = 0 of order 2q. As previously outlined, the matrix Cm (f0 ) is singular, so that the solution of the rank-1 corrected system is considered, whose matrix is given by Cm (f0 ) + (f0 (π/m)/m)eeT , with e = [1, . . . , 1]T . Since the position of the zero x0 = 0 at the coarser levels is never shifted, then the function ps (x) = [2 − 2 cos(π − x)]r in the projectors is the same at all the subsequent levels s. To test TGM/MGM linear convergence with rate independent of the size m0 we tried for diﬀerent r: according to (3.6a), we must choose r at least equal to 1 if q = 1 and at least equal to 2 if q = 2, 3, while according to (4.3a) we must always choose r equal to q. In Table 1, we report the number of iterations required for convergence. As expected, it results to be bounded by a constant irrespective of the problem size. Notice also that, in general, the MGM requires the same number of iterations than the TGM. In other words, the approximation due to the Coarse Grid Correction in the V-cycle does not introduce any signiﬁcant loss of accuracy. By using tensor arguments, the previous results plainly extend to the multilevel case. In Table 2 we consider the case of generating function f0 (x, y) = f0 (x) + f0 (y), that arises in the uniform ﬁnite diﬀerence discretization of elliptic constant coeﬃcient diﬀerential equations on a square with Neumann boundary conditions, see, e.g., [24].

392

C. Tablino Possio Table 1. Number of required iterations – 1D Case: f0 (x) = [2 − 2 cos(x)]q and p(x) = [2 − 2 cos(π − x)]r .

m0

q=1 r=1

16 32 64 128 256 512

7 7 7 7 7 7

TGM q=2 r=1 r=2 15 16 16 16 16 16

13 15 16 16 16 16

q=3 r=2 r=3 28 34 35 35 35 35

24 32 34 35 35 35

m0

q=1 r=1

16 32 64 128 256 512

1 7 7 7 7 7

MGM q=2 r=1 r=2 1 16 17 18 18 18

1 15 16 16 16 16

q=3 r=2 r=3 1 34 35 35 35 35

1 32 34 35 35 35

Table 2. Number of required iterations – 2D Case: f0 (x, y) = [2 − 2 cos(x)]q + [2 − 2 cos(y)]q and p(x, y) = [2 − 2 cos(π − x)]r + [2 − 2 cos(π − y)]r .

m0

q=1 r=1

162 322 642 1282 2562 5122

15 16 16 16 16 16

TGM q=2 r=1 r=2 34 36 36 36 36 36

30 35 36 36 36 36

q=3 r=2 r=3 – 71 74 74 74 74

– 67 73 73 73 73

m0

q=1 r=1

162 322 642 1282 2562 5122

1 16 16 16 16 16

MGM q=2 r=1 r=2 1 36 36 36 37 37

1 35 36 36 36 36

q=3 r=2 r=3 1 71 74 74 74 74

1 67 73 73 73 73

5.2. Case x0 = π (integral-like problems) DCT-III matrices Am0 = Cm0 (f0 ) whose generating function shows a unique zero at x0 = π are encountered in solving integral equations, for instance in image restoration problems with Neumann (reﬂecting) boundary conditions [18]. According to Lemma 3.2, if x0 = π, then the generating function f1 of the coarser matrix Am1 = Cm1 (f1 ), m1 = m0 /2, has a unique zero at 0, whose order equals the order of x0 = π with respect to f0 plus two. It is worth stressing that in such a case the projector at the ﬁrst level is singular so that its rank-1 Strang correction is considered. This choice gives rise in a natural way to the rank-1 correction considered in Section 5.1. Moreover, starting from the second coarser level, the new location of the zero is never shifted from 0. In Table 3 the number of iterations required for convergence, both in the one-level and two-level case, is reported.

6. Computational costs and conclusions Some remarks about the computational costs are required in order to highlight the optimality of the proposed procedure. s is banded, the Since the matrix Cms (p) appearing in the deﬁnition of Ps+1 s cost of a matrix vector product involving Ps+1 is O(ms ). Therefore, the ﬁrst con-

V-cycle Optimal Convergence for DCT-III Matrices

393

Table 3. Number of required iterations – 1D Case: f0 (x) = 2 + 2 cos(x) and p0 (x) = 2 − 2 cos(π − x) and 2D Case: f0 (x, y) = 4 + 2 cos(x) + 2 cos(y) and p0 (x, y) = 4 − 2 cos(π − x) − 2 cos(π − y). m0

TGM

MGM

m0

TGM

MGM

16 32 64 128 256 512

15 14 12 11 10 8

1 14 13 13 12 10

162 322 642 1282 2562 5122

7 7 7 7 7 7

1 7 7 6 6 6

dition in Deﬁnition 1.1 is satisﬁed. In addition, notice that the matrices at every level (except for the coarsest) are never formed since we need only to store the O(1) nonzero Fourier coeﬃcients of the related generating function at each level for matrix-vector multiplications. Thus, the memory requirements are also very low. With respect to the second condition in Deﬁnition 1.1 we stress that the representation of Ams+1 = Cms+1 (fs+1 ) can be obtained formally in O(1) operations by virtue of (3.5). In addition, the zeros of fs+1 and their orders are obtained according to Lemma 3.2 by knowing the zeros of fs and their orders. Furthermore, each iteration of TGM costs O(m0 ) operations as Am0 is banded. In conclusion, each iteration of the proposed TGM requires O(m0 ) operations. With regard to MGM, optimality is reached since we have proven that there exists δ independent from both m and smin so that the number of required iterations results uniformly bounded by a constant irrespective of the problem size. In addition, since each iteration has a computational cost proportional to matrixvector product, Deﬁnition 1.1 states that such a kind of MGM is optimal. Finally, we report some remarks about the performances of the proposed method with respect to those achieved by considering Fast Cosine Transform (FCT) standard resolutions [27]. Let us consider the linear system Am xm = bm , where Am belongs to the DCT-III matrix algebra. Then, the solution is given by T xm = Qm Λ−1 m Qm b m ,

(6.1)

where Λm is the diagonal matrix holding the eigenvalues of Am . Since, these eigenvalues can be evaluates as [QT Am e1 ]i [Λm ](i,i) = m T , i = 1, . . . , m, [Qm e1 ]i with e1 = [1, 0, . . . , 0]T , the solution (6.1) can be obtained in three FCT, i.e., within O(m log m) operations. Thus, such a computational cost is not substantially higher than the cost of the proposed MGM. In addition, the MGM implementation is a delicate task since it clearly involves an eﬃcient memory allocation, both with respect to the recursion itself and the data stored in the setup phase. Nevertheless, even in the one-level case, for increasing dimensions, the solution computed by means of FCT can show a worsening in the accuracy greater

394

C. Tablino Possio Table 4. Relative error in Euclidean norm er in the case of our MGM Matlab code and FCT built-in Matlab function based code – 1D case: f0 (x) = [2 − 2 cos(x)]2 (p(x) = [2 − 2 cos(π − x)]2 and stopping criterion equal to 10−7 for the MGM). m0

MGM er

FCT er

16 32 64 128 256 512 1024 2048 4096

2.23e-011 3.99e-010 3.20e-010 2.43e-008 2.68e-007 4.46e-006 5.47e-005 5.67e-004 9.22e-003

8.43e-014 1.87e-013 1.16e-011 2.68e-010 3.54e-009 1.04e-007 1.61e-006 1.72e-002 3.93e-001

than the solution computed by MGM, as, for instance, in the case of the generating function f0 (x) = 2 − 2 cos(x). The loss of accuracy appears even more suddenly in the case of f0 (x) = [2 − 2 cos(x)]2 : for a size larger than 2000, the results delivered by our MGM implementation are deﬁnitely more accurate than those obtained by using the Matlab FCT code and show a more predictable pattern. More precisely, for increasing dimensions, Table 4 compares the relative error in Euclidean norm obtained by considering our MGM Matlab implementation and by using the builtin Matlab dct function. For the sake of completeness, we have to mention that our MGM CPU times are larger than those of the FCT code. Nevertheless, we think that the comparison is unfair since the former refer to a non-optimized Matlab code (the matrix-vector product key operation is implemented in O(m) operations by using loops, but it is known that loops slow down the Matlab performances), while the latter make use of a built-in executable function. Moreover, the MGM, due to its iterative nature, has also a greater ﬂexibility related to the choice of threshold in the stopping criterion, whenever the considered application does not require a high accuracy. We recall that this situation is not academic and indeed it does occur when considering image restoration problems [18, 17], where we cannot expect an error less than the noise level. As a conclusion, we observe that the reported numerical tests in Section 5 show that the requirements on the order of zero in the projector could be weakened. Future works will deal with this topic and with the extension of the convergence analysis in the case of a general location of the zeros of the generating function. Acknowledgment Warm thanks to the referee for very pertinent and useful remarks.

V-cycle Optimal Convergence for DCT-III Matrices

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References [1] A. Aric` o, M. Donatelli, A V-cycle multigrid for multilevel matrix algebras: proof of optimality. Numer. Math. 105 (2007), no. 4, 511–547 (DOI 10.1007/s00211-006-0049-7). [2] A. Aric` o, M. Donatelli, S. Serra-Capizzano, V-cycle optimal convergence for certain (multilevel) structured linear systems. SIAM J. Matrix Anal. Appl. 26 (2004), no. 1, 186–214. [3] O. Axelsson, M. Neytcheva, The algebraic multilevel iteration methods – theory and applications. In Proceedings of the Second International Colloquium on Numerical Analysis (Plovdiv, 1993), 13–23, VSP, 1994. [4] R.H. Chan, T.F. Chan, C. Wong, Cosine transform based preconditioners for total variation minimization problems in image processing. In Iterative Methods in Linear Algebra, II, V3, IMACS Series in Computational and Applied Mathematics, Proceedings of the Second IMACS International Symposium on Iterative Methods in Linear Algebra, Bulgaria, 1995, 311–329. [5] R.H. Chan, M. Donatelli, S. Serra-Capizzano, C. Tablino-Possio, Application of multigrid techniques to image restoration problems. In Proceedings of SPIE-Session: Advanced Signal Processing: Algorithms, Architectures, and Implementations XII, Vol. 4791 (2002), F. Luk Ed., 210–221. [6] R.H. Chan, M.K. Ng, Conjugate gradient methods for Toeplitz systems. SIAM Rev. 38 (1996), no. 3, 427–482. [7] R.H. Chan, S. Serra-Capizzano, C. Tablino-Possio, Two-grid methods for banded linear systems from DCT III algebra. Numer. Linear Algebra Appl. 12 (2005), no. 2-3, 241–249. [8] G. Fiorentino, S. Serra-Capizzano, Multigrid methods for Toeplitz matrices. Calcolo 28 (1991), no. 3-4, 283–305. [9] G. Fiorentino, S. Serra-Capizzano, Multigrid methods for symmetric positive deﬁnite block Toeplitz matrices with nonnegative generating functions. SIAM J. Sci. Comput. 17 (1996), no. 5, 1068–1081. [10] R. Fischer, T. Huckle, Multigrid methods for anisotropic BTTB systems. Linear Algebra Appl. 417 (2006), no. 2-3, 314–334. [11] W. Hackbusch, Multigrid methods and applications. Springer Series in Computational Mathematics, 4. Springer-Verlag, 1985. [12] T. Huckle, J. Staudacher, Multigrid preconditioning and Toeplitz matrices. Electron. Trans. Numer. Anal. 13 (2002), 81–105. [13] T. Huckle, J. Staudacher, Multigrid methods for block Toeplitz matrices with small size blocks. BIT 46 (2006), no. 1, 61–83. [14] X.Q. Jin, Developments and applications of block Toeplitz iterative solvers. Combinatorics and Computer Science, 2. Kluwer Academic Publishers Group, Dordrecht; Science Press, Beijing, 2002. [15] D.G. Luenberger, Introduction to Dynamic Systems: Theory, Models, and Applications, John Wiley & Sons Inc., 1979. [16] M.K. Ng, Iterative methods for Toeplitz systems. Numerical Mathematics and Scientiﬁc Computation. Oxford University Press, 2004.

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Operator Theory: Advances and Applications, Vol. 199, 397–419 c 2010 Birkh¨ auser Verlag Basel/Switzerland

The Ratio Between the Toeplitz and the Unstructured Condition Number Siegfried M. Rump and Hiroshi Sekigawa Abstract. Recently it was shown that the ratio between the normwise Toeplitz structured condition number of a linear system and the general unstructured condition number has a ﬁnite lower bound. However, the bound was not explicit, and nothing was known about the quality of the bound. In this note we derive an explicit lower bound only depending on the dimension n, and we show that this bound is almost sharp for all n. Mathematics Subject Classiﬁcation (2000). 15A12, 26D05. Keywords. Structured condition number, Toeplitz matrix, Mahler measure, polynomial norms.

1. Notation and problem formulation For a system of linear equations Ax = b with A ∈ Rn×n , x, b ∈ Rn , the condition number characterizes the sensitivity of the solution x with respect to inﬁnitely small perturbations of the matrix A. For ε > 0, denote Mε := Mε (A) := {ΔA ∈ Rn×n : ΔA ≤ εA},

(1.1)

where throughout the paper · denotes the spectral norm for matrices and for vectors. Denote by Pε := Pε (A, x) the set of all vectors Δx ∈ Rn for which there exists ΔA ∈ Mε with (A + ΔA)(x + Δx) = Ax. Then the (unstructured) normwise condition number is deﬁned by κ(A, x) := lim sup

ε→0 Δx∈Pε

Δx . εx

(1.2)

It is well known that κ(A, x) = A−1 A, such that the (unstructured) condition number does not depend on x. If the matrix A has some structure, it seems reasonable to restrict the set Mε to matrices with similar structure. For a = (a−(n−1) , . . . , a−1, a0 , a1 , . . . , an−1 ), the

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S.M. Rump and H. Sekigawa

n × n Toeplitz matrix Tn (a) is of the form ⎛ a0 a1 . . . ⎜ .. ⎜ a−1 . a0 ⎜ T := Tn (a) := ⎜ .. . . .. .. ⎝ . a−(n−1) . . . a−1

⎞ an−1 .. ⎟ . ⎟ ⎟ ∈ Rn×n . ⎟ a1 ⎠ a0

(1.3)

For given (nonsingular) Toeplitz matrix T , restricting Mε to Toeplitz matrices changes (1.2) into the Toeplitz condition number κToep (T, x) [10], [14], [2, Section 13.3]. Since the set of perturbations ΔA is restricted, it follows κToep (T, x) ≤ κ(T, x) = T −1 T . Note that in contrast to the general condition number, the Toeplitz condition number depends on x. However, there exists always a worst case x such that both condition numbers coincide [14, Theorem 4.1]: A B sup κToep (T, x) = T −1 T . x =0

5 In [14, Theorem 10.2] it was shown that κToep (T, x) ≥ 2−1/2 κ(T, x) (see also [2, Theorem 13.14]), hence the ratio κToep /κ is bounded below by [2T −1 T ]−1/2. The question arises, how small can the Toeplitz condition number actually be compared to the general condition number? In a recent survey paper on Toeplitz and Hankel matrices [4], B¨ ottcher and Rost note “One expects that κToep (T, x) is in general signiﬁcantly smaller than κ(T, x), but, curiously up to now no convincing example in this direction is known.” Furthermore, B¨ottcher and Rost continue to note that, as proved in [3] (submitted in 2002 but appeared in 2005), it seems rather hopeless to ﬁnd examples numerically (see also [2, Theorem 13.20]): Theorem 1.1 (B¨ottcher, Grudsky, 2002). Let x0 , x1 , . . . , xn−1 ∈ C be independent random variables whose real and imaginary parts are subject to the standard normal distribution and put x = (xj )n−1 j=0 . There are universal constants δ ∈ (0, ∞) and n0 ∈ N such that Toep

(Tn (a), x) κ 99 δ Probability ≥ 3/2 > κ(Tn (a), x) 100 n for all ﬁnitely supported sequences a and all n ≥ n0 . Notice that generically κ(Tn (a), x) remains bounded or increases exponentially fast as n goes to inﬁnity. Since in the case of exponential growth the factor δ/n3/2 is harmless, it follows that with high probability that κToep (Tn (a), x) increases exponentially fast together with κ(Tn (a), x). In [14] the ﬁrst author showed a lower bound on the ratio κToep /κ which surprisingly depends only on the solution x, not on A (see also [2, Theorem 13.16]). However, despite some examples of small dimension (inspired by Heinig [9]) no general examples could be derived.

The Toeplitz and the Unstructured Condition Number

399

In this note we 1. derive a general lower bound on κToep (T, x)/κ(T, x) only depending on the dimension n, and 2. show that this lower bound is almost sharp for all n. The solution of both problems is based on the minimization of the smallest singular value of a class of Toeplitz matrices (2.2) and its surprising connection to a lower bound on the coeﬃcients of the product of two polynomials. We will prove in Corollary 2.11 that √ 2 2n κToep(A, x) n×n n : A∈R ≥ inf{ Toeplitz, 0 = x ∈ R } > , n−1 n−1 Δ κ(A, x) nΔ where Δ = 3.209912 . . .. We denote by σmin (A) the smallest singular value of the matrix A, and by J the permutation matrix (“ﬂip matrix”) mapping (1, . . . , n) into (n, . . . , 1).

2. Main results Let a linear system Ax = b with Toeplitz matrix A be given. The deﬁning equation (A + ΔA)(x + Δx) = Ax with ΔA ≤ εA implies Δx = −A−1 ΔAx + O(ε).

(2.1)

For Toeplitz perturbations, we have ΔA = T (Δa) with Δa ∈ R according to (1.3), and using ideas from [10] a computation shows [14] ⎛ ⎞ x1 x2 . . . xn ⎜ ⎟ x1 x2 . . . xn ⎟ ∈ Rn×(2n−1) . ΔAx = JΨx Δa with Ψx := ⎜ ⎝ ⎠ ... x1 x2 . . . xn (2.2) In [14, Lemma 6.3] it was shown that the spectral matrix norm of ΔA and Euclidean norm of Δa are related by √ 1 √ ΔA ≤ Δa ≤ 2ΔA. (2.3) n 2n−1

Combining this with the deﬁnition of κToep (A, x) and (2.1) yields [14, Theorem 6.5] √ A−1 JΨx A 1 (2.4) with √ ≤ γ ≤ 2, κToep (A, x) = γ x n so that A−1 JΨx ≥ A−1 σmin (Ψx ) implies [14, Corollary 6.6] κToep (A, x) 1 A−1 JΨx 1 σmin (Ψx ) ≥√ ≥ √ . κ(A, x) n A−1 x n x

(2.5)

Surprisingly, this lower bound depends only on the solution x. That means, a given solution x implies a lower bound for κToep (A, x)/κ(A, x) for any Toeplitz matrix A.

400

S.M. Rump and H. Sekigawa

We will show that the lower bound in (2.5) is achievable up to a small factor. For this we ﬁrst construct for given x a Toeplitz matrix A with ratio κToep /κ near σmin (Ψx )/x. Let ﬁxed but arbitrary x ∈ Rn be given. For simplicity assume x = 1. First we will show that for δ > 0 there exists a Toeplitz matrix A with A−1 JΨx < A−1 σmin (Ψx ) + δ. Denote by y ∈ Rn , y = 1, a left singular vector of Ψx to σmin (Ψx ), so that T y Ψx = σmin (Ψx ). By Ψx ΨTx = JΨx ΨTx J T we may assume either y = Jy or y = −Jy. Deﬁne by ⎛ ⎞ p1 ⎜ ⎟ ⎜ p2 . . . ⎟ ⎟ ∈ Rn×n , L(p1 , . . . , pn ) := ⎜ ⎜ . .. ⎟ . ⎝ .. ⎠ . .. pn . . . p2 p1 a lower triangular Toeplitz matrix depending on p ∈ Rn . Deﬁne B := L(y1 , y2 , . . . , yn )

and C := L(0, yn , yn−1, . . . , y2 )

and Rε := (B + εI)(B + εI)T − CC T .

(2.6)

If Rε is invertible, then the Gohberg-Semencul formula ([8], see also [4, Th. 3.3]) implies that Aε := Rε−1 is a (symmetric) Toeplitz matrix. Furthermore, a direct computation using y = ±Jy yields R0 = yy T

(2.7)

which implies det(R0 ) = 0 for n ≥ 2. The determinant of Rε is a monic polynomial of degree 2n in ε, thus Rε is nonsingular for all 0 = ε < ε0 for small enough ε0 . Hence there is a constant α, independent of ε, with Rε Ψx ≤ yy T Ψx + αε = σmin (Ψx ) + αε, 2 (Ψx ) is the only nonzero eigenvalue of the latter equality because σmin

yy T Ψx (yy T Ψx )T . Since Aε = Rε−1 is a Toeplitz matrix and y = ±Jy, (2.4) implies the following result, which is trivially also true for n = 1. Theorem 2.1. Let 0 = x ∈ Rn be given. Then for all δ > 0 there exists a Toeplitz matrix A ∈ Rn×n with A−1 σmin (Ψx ) ≤ A−1 JΨx < A−1 σmin (Ψx ) + δ and κToep (A, x) = γ · κ(A, x)

σmin (Ψx ) + δ x

for

√ 1 √ ≤γ≤ 2 n

and

0 ≤ δ <

√

2δ.

The Toeplitz and the Unstructured Condition Number

401

For x = 0, the matrix Ψx has full rank because otherwise each n×n submatrix of Ψx would be singular, taking the leftmost submatrix in Ψx would imply x1 = 0, the second leftmost would imply x2 = 0 and so forth. Thus σmin (Ψx ) = min σmin (Ψx ) > 0 0 =x∈R x x=1

μn := min

(2.8)

for all n, and Theorem 2.1 yields Corollary 2.2. For all n, √ κToep (A, x) 1 : A ∈ Rn×n Toeplitz, 0 = x ∈ Rn } ≥ √ μn . 2μn ≥ inf{ κ(A, x) n In the remaining of the paper we will estimate μn to characterize the inﬁmum of κToep /κ. The matrix Ψx is also known as “polynomial matrix” 1 . Identifying n−1 a vector x = (x1 , . . . , xn ) ∈ Rn with the polynomial x(t) := xn−ν tν ∈ R[t], a ν=0

little computation yields z = y T Ψx

⇔

z(t) = y(t)x(t),

(2.9)

and therefore (2.10) y T Ψx = xT Ψy . This, of course, can also be veriﬁed by direct computation. We deﬁne the norm x(t) of a polynomial by the norm x of its coeﬃcient vector. Since y T Ψx = σmin (Ψx ), we can characterize μn by μn = min{P Q : P, Q ∈ R[t], deg(P ) = deg(Q) = n − 1, P = Q = 1}. (2.11) To give lower and upper bounds for μn , we ﬁrst describe some related results for polynomials. The supremum norm P E of a complex univariate polynomial P on a compact set E ⊂ C is deﬁned as P E := sup |P (z)|.

(2.12)

z∈E

In [12] Kneser gave the exact lower bound for the supremum norm on the interval [−1, 1] of the product of two polynomials. Theorem 2.3 (Kneser, 1934). Suppose that P Q = R, where P , Q and R are complex polynomials of degree m, n − m and n, respectively. Then for all m and n P [−1,1] Q[−1,1] ≤ Km,n R[−1,1] , where Km,n := 2

n−1

n−m

m (2k − 1)π (2k − 1)π 1 + cos 1 + cos . 2n 2n

k=1

k=1

This bound is exactly attained by the Chebyshev polynomial of degree n. 1 Many

thanks to Ludwig Elsner, Bielefeld, for pointing to this connection.

402

S.M. Rump and H. Sekigawa

To estimate μn , we need similar results for the unit disc D. Boyd’s result in [5, 6] gives a sharp inequality for this case. To describe Boyd’s results, we deﬁne the Mahler measure. For a complex polynomial F in k variables the Mahler measure of F is deﬁned as 4 1

4 1 √ √ 2π −1t1 2π −1tk M (F ) := exp ··· log |F (e ,...,e )|dt1 · · · dtk . 0

0

Theorem 2.4 (Boyd, 1992/94). Let R be a polynomial of degree n with complex coeﬃcients and suppose that P Q = R. Then for the norm · D as in (2.12) on the unit disc D P D QD ≤ δ n RD , where δ = M (1 + x + y − xy) = 1.7916228 . . .. The constant is best possible. As written in section 3 of [5], the constant δ can be expressed in terms of Clausen’s integral

4 θ ∞ sin kθ t dt = Cl2 (θ) = − log 2 sin , 2 k2 0 k=1

or, in terms of I(θ), where

4

θ

I(θ) = 0

t dt = Cl2 (π − θ). log 2 cos 2

Using Catalan’s constant G = I(π/2) = Cl2 (π/2) ≈ 0.9160, we can write δ = e2G/π . Theorem 2.4 implies a lower bound for μn . To obtain an upper bound for μn , we estimate the supremum norms of the following polynomials. Deﬁne Fn (t) as t2n + (−1)n . Let Pˆn (t) be the monic polynomial of degree n with the zeros ˆ n (t) be the monic polynomial of degree n of Fn (t) in the right half-plane, and Q ˆ n = Fn and Q ˆ n (t) = with the zeros of Fn (t) in the left half-plane. It follows Pˆn Q nˆ (−1) Pn (−t). Lemma 2.5. For the norm · D as in (2.12) on the unit disc D, the following inequalities hold true. π ˆ n D = Q ˆ n (1) > δ n . e 8n δ n > Pˆn D = (−1)n Pˆn (−1) = Q ˆ n (1)2 , where Kp,q is the constant in Remark 2.6. When n is even, 2Kn/2,n = Q Theorem 2.3. Combining Theorem 2.4 and Lemma 2.5, where the proof of the latter is deferred to the appendix, with (2.11), we obtain an upper and a lower bound for μn . Before we state our ﬁnal result, we prove that we may assume without loss of generality that polynomials P and Q minimizing μn as in (2.11) must both have all their roots on the unit circle. This is also useful to identify such polynomials P and Q numerically for small n. In fact, the following Theorem 2.7 shows more, namely that for ﬁxed (normed) Q there is a (normed) polynomial P with only roots on the unit circle and minimizing P Q.

The Toeplitz and the Unstructured Condition Number

403

Theorem 2.7. For two nonzero real univariate polynomials P and Q with P = Q = 1, there exists a real univariate polynomial P such that deg(P ) = deg(P ), P = 1, all zeros of P lie on the unit circle and P Q ≤ P Q. The proof of Theorem 2.7 is rather involved, and thus deferred to the appendix. An immediate consequence is the following corollary. Corollary 2.8. μn

=

min{P Q : P, Q ∈ R[t], deg(P ) = deg(Q) = n − 1, P = Q = 1, and P , Q have all zeros on the unit circle}.

Now we can prove the following upper and lower bounds for μn . Theorem 2.9.

√ 2 2(n + 1) > μn+1 ≥ √ , Δn 2n + 1Δn

where Δ := e4G/π for Catalan’s constant G. It is Δ = δ 2 , where δ is the constant in Theorem 2.4. Note that Δ = 3.209912 . . . . Remark 2.10. Using Proposition 2.12 at the end of this section, we can improve the upper bound to √ C n+1 , Δn where C is a constant independent of n. n Proof. Let F be a complex polynomial ν=0 aν tν . Then, the following inequalities among norms of F hold. √ n + 1F ≥ |F |1 ≥ F D ≥ F . (2.13) n Here, |F |1 is deﬁned as ν=0 |aν |. Real polynomials P and Q minimizing μn have all their roots on the unit circle. For this case the right-most inequality in (2.13) improves into √ (2.14) F D ≥ 2F which follows from a much more general result2 in [16], see also [17, (7.71)]. From Theorem 2.4, for real polynomials P and Q of degree n, we have P QD 1 1 ≥ 2n = n . P D QD δ Δ Therefore, for polynomials P and Q with P = Q = 1, the inequalities P Q ≥ √

2 2P QD ≥√ 2n + 1P D QD 2n + 1Δn

follow from (2.13) and (2.14). This proves the lower bound for μn+1 . 2 Thanks

to P. Batra, Hamburg, for pointing to this reference.

404

S.M. Rump and H. Sekigawa

ˆ n be as in Lemma 2.5. An upper bound for Pˆn Q ˆ n ) ˆ n /(Pˆn Q Let Pˆn and Q √ ˆ ˆ is an upper bound for μn+1 . Since Pn Qn = 2, the inequalities √ √ ˆ n 2 2(n + 1) Pˆn Q ≤ < √ √ ˆ n ˆ n D / n + 1) Δn Pˆn Q (Pˆn D / n + 1)(Q

follow from (2.13) and Lemma 2.5.

Inserting this into Corollary 2.2 characterizes the asymptotic behavior of the worst ratio between the unstructured and structured condition number for Toeplitz matrices. Corollary 2.11. For all n, √ 2 Toep 8 2n κ (A, x) 2 n×n n : A∈R > inf Toeplitz, 0 = x ∈ R , (2.15) > Δn−1 κ(A, x) nΔn−1 where Δ = 3.209912 . . . is the constant in Theorem 2.9. We can improve the upper bound using the following proposition, the proof of which is given in the Appendix. Proposition 2.12. ˆ n ||n1/4 ||Pˆn ||n1/4 ||Q 1 = lim =√ . ˆn ||D ˆ n ||D n→∞ ||P n→∞ ||Q 2 lim

By similar arguments of the proof for Theorem 2.9, we obtain the following improved upper bound. Corollary 2.13. There exists a constant C > 0 such that for all n, 2 Toep 8 √ κ C n (A, x) n×n n : A∈R > inf Toeplitz, 0 = x ∈ R , Δn−1 κ(A, x) where Δ = 3.209912 . . . is the constant in Theorem 2.9.

3. Approximation of μn Next we show how to approximate Ψx ∈ Rn×(2n−1) minimizing σmin (Ψx ). Using Ψx , a Toeplitz matrix with small ratio κToep /κ can be constructed following the discussion preceding Theorem 2.1. For given unit vector x ∈ Rn and x(t) := n−1 xn−ν tν deﬁne Ψx as in (2.2), and let y ∈ Rn be a unit left singular vector to ν=0

σmin (Ψx ) of Ψx . With y(t) :=

n−1

yn−ν tν as in the discussion following Corollary

ν=0

2.2 we have x = y = x(t) = y(t) = 1

and x(t)y(t) = y T Ψx = σmin (Ψx ).

The Toeplitz and the Unstructured Condition Number n

approximate μn

2 3 4 5 6 7 8

0.70710678118655 0.33333333333333 0.13201959446019 0.04836936580270 0.01702151213258 0.00584679996238 0.00197621751074

rigorous bounds of μn [ [ [ [ [ [ [

0.70710678118 0.33333333333 0.13201959446 0.04836936580 0.01702151213 0.00584679996 0.00197621751

, , , , , , ,

0.70710678119 0.33333333334 0.13201959447 0.04836936581 0.01702151214 0.00584679997 0.00197621752

μ ˆn = ] ] ] ] ] ] ]

405

ˆn Q ˆ n || ||P ˆn ||||Q ˆ n || ||P

0.707107 0.353553 0.141421 0.051777 0.018183 0.006234 0.002104

μ ˆn /μn 1.0000 1.0607 1.0712 1.0705 1.0682 1.0662 1.0647

Table 1

For ﬁxed x(t), the polynomial y(t) minimizes x(t)y(t) subject to y(t) = 1. Now (2.10) implies xT Ψy = σmin (Ψx ) and therefore σmin (Ψy ) ≤ σmin (Ψx ). Iterating the process, that is replacing x by y and computing y as a left singular vector to σmin (Ψx ), generates a monotonically decreasing and therefore convergent sequence. Practical experience suggests that for generic starting vector x this sequence converges mostly to the same limit, presumably μn . In any case this limit is an upper bound for μn . Table 1 displays this limit for some values of n. To ensure that the limit is not a local but the global minimum min σmin (Ψx ), x

a veriﬁed global optimization method was used [13] for computing rigorous lower and upper bounds for μn . Such methods take all procedural, approximation and rounding errors into account and are, provided the computer system works to its speciﬁcations, rigorous (see, for example, [7]). For given n and using (2.11) this means 2n variables. This was possible up to n = 5 with reasonable eﬀort. The rightmost column in Table 1 displays the computed bounds for μn . For larger values of n, the number of variables was signiﬁcantly reduced using Theorem 2.7. Since minimizers P, Q have only roots on the unit circle it follows P (z) = ±z n P (1/z) and similarly for Q, i.e., the coeﬃcient vectors are (skew-)symmetric to reﬂection. Using this allows the computation of rigorous bounds for μn up to n = 8 with moderate eﬀort.3 The best-known lower and upper bounds for μn are by Kaltofen et al. [11]. They compute veriﬁed lower bounds until n = 18. For bounding μ18 from below they need 25 days of computing time. Results and more background are summarized in [15]. Computational evidence supports the following conjecture. Conjecture 3.1. There are polynomials P, Q ∈ R[t] with deg P = deg Q = n − 1 and P = Q = 1 with μn = P Q such that all coeﬃcients of P are positive, Q(t) = P (−t) and all roots of P and Q lie on the unit circle. The roots aν ± ibν of P have all positive real parts aν , and the roots of Q are −aν ± ibν . 3 Thanks to Kyoko Makino for performing the veriﬁed global optimization using the COSYpackage [1].

406

S.M. Rump and H. Sekigawa

ˆ n || ˆn Q ||P ˆ ˆ Finally, the values μ ˆn = ||P ˆn ||||Q ˆ n || for the polynomials Pn , Qn as in Lemma 2.5 ˆ n are not far from the and the ratio μ ˆ n /μn is displayed as well. It seems that Pˆn , Q optimum. This is supported by Proposition 2.12.

4. Appendix Proof of Lemma 2.5. Since we can write ⎧ n−1

⎪ 2 ⎪ kπ ⎪ 2 ⎪ (t + 1) t + 1 , if n is odd, + 2t cos ⎪ ⎨ n k=1 ˆ Qn (t) = n

⎪ 2 ⎪ (2k − 1)π ⎪ 2 ⎪ t + 1 , if n is even, + 2t cos ⎪ ⎩ 2n

(4.1)

k=1

(2k−1)π ˆ n D = Q ˆ n (1). and cos kπ > 0 for the values of k in question, we have Q n , cos 2n ˆ n , we have Q ˆ n D = Pˆn D and Q ˆ n (1) = (−1)n Pˆn (−1). From the deﬁnition of Q First we prove the inequalities in Lemma 2.5 when n is odd. From (4.1), we have n−1

2 n+1 kπ ˆ . 1 + cos Qn (1) = 2 2 n k=1

Therefore, n−1

2 (n + 1) log 2 kπ ˆ n (1) = log Q + . log 1 + cos 2 n

k=1

Let a and b be real numbers with a < b. For a real function f such that f ≤ 0 on the interval [a, b], we have

4 b f (a) + f (b) a+b f (x)dx ≥ (b − a) (b − a)f ≥ . (4.2) 2 2 a 1 ], [ 2k−1 , 2k+1 ] (k = 1, 2, Applying (4.2) to f = log(1 + cos πx) on intervals [0, 2n 2n 2n n−1 k k+1 . . . , 2 ) for an upper estimation, and on intervals [ n , n ] (k = 0, 1, . . . , n−3 2 ), 1 [ n−1 , ] for a lower estimation, we have 2n 2

ˆ n (1) (n + 1) log 2 1 π log Q − + log 1 + cos n 2n 2n 4n 4 12 log(1 + cos πx)dx ≥ 0

ˆ n (1) log 2 log Q 1 (n − 1)π ≥ − − log 1 + cos . n 2 4n 2n

The Toeplitz and the Unstructured Condition Number πx 2 ,

Since 1 + cos πx = 2 cos2 4

1 2

4

it follows

πx log 2 cos2 dx 2 0

4 π2 log 2 log 2 2 t dt − = log δ − . = log 2 cos π 0 2 2 2

log(1 + cos πx)dx =

0

407

1 2

(4.3)

From (4.3), we have ˆ n (1) log 2 log Q 1 π − + log 1 + cos n 2n 2n 4n

ˆ n (1) log Q 1 (n − 1)π ≥ log δ ≥ − log 1 + cos . n 4n 2n Therefore, the following inequalities hold.

(n − 1)π 1 ˆ n (1) ≥ log Q n log δ + log 1 + cos 4 2n log 2 1 π ≥ n log δ + − log 1 + cos . 2 2 4n Since

(n − 1)π log 1 + cos 2n

(4.4)

π π π = log 1 + sin < log 1 + < , 2n 2n 2n

we can estimate the left-hand side of (4.4) by

1 (n − 1)π π n log δ + log 1 + cos < n log δ + . 4 2n 8n An estimation for the right-hand side of (4.4) is as follows. π Since log 2 > log 1 + cos 4n , we have n log δ +

π log 2 1 − log 1 + cos > n log δ, 2 2 4n

and therefore

π ˆ n (1) > n log δ > log Q 8n proves Lemma 2.5 for odd n. When n is even, we have n log δ +

n

2 n log 2 (2k − 1)π ˆ n (1) = log Q . + log 1 + cos 2 2n

k=1

n Applying (4.2) to f = log(1 + cos πx) on intervals [ nk , k+1 n ] (k = 0, 1, . . . , 2 − 1), n−1 1 1 2k−1 2k+1 [ 2n , 2 ] for an upper estimation, and on intervals [0, 2n ], [ 2n , 2n ] (k = 1, 2,

408 ...,

S.M. Rump and H. Sekigawa n 2

1 − 1), [ n−1 2n , 2 ] for a lower estimation, we have ˆ n (1) log 2 4 12 log Q log (1 + cos πx) dx − ≥ n 2 0 ˆ n (1) log 2 log Q 1 π ≥ − − log 1 + cos n 2 2n

2n 1 (n − 1)π log 2 − log 1 + cos . + 2n 4n 2n

Therefore, the inequalities

π log 2 1 (n − 1)π 1 − + log 1 + cos n log δ + log 1 + cos 2 2n 2 4 2n ˆ ≥ log Qn (1) ≥ n log δ hold, and from similar arguments for odd n, the inequalities π ˆ n (1) > n log δ n log δ + > log Q 8n prove Lemma 2.5 for even n.

To prove Theorem 2.7, we need the following lemmas, corollaries and algorithm. Lemma 4.1. Let F and G be nonzero complex univariate polynomials, and ζ be a ﬁxed complex number on the unit circle. Deﬁne ν : R → R by ν(r) :=

(t − rζ)F . (t − rζ)G

Then, the following statements hold. 1. ν(r) has a minimum at either r = 1 or −1. 2. If ν(1) is not a minimum, then ν(r) > ν(0) for any r > 0. Proof. Since ν(r) is nonnegative, it is suﬃcient to prove that N (r) = ν(r)2 has the above properties. For P (t) = nk=0 ak tk , we have n n 2 2 2 ak−1 ak + ζ ak−1 ak r + P 2 . (4.5) (t − rζ)P = P r − ζ k=1

k=1

Therefore, we can write (t − rζ)F 2

= f1 r2 + f2 r + f1 ,

(t − rζ)G

= g1 r 2 + g2 r + g 1 ,

2

where f1 = F 2 , g1 = G2 , and f2 , g2 are real numbers. Therefore, we have N (r) =

(f1 g2 − f2 g1 )(r2 − 1) . (t − rζ)G2

If f1 g2 − f2 g1 = 0, then N (r) is constant and the statements are clear.

The Toeplitz and the Unstructured Condition Number

409

If f1 g2 − f2 g1 > 0, then N (r) tends to f1 /g1 = N (0), as r tends to ±∞. Furthermore, N (r) is monotonically increasing on (−∞, −1], monotonically decreasing on [−1, 1] and monotonically increasing on [1, ∞). Therefore, N (r) has a minimum at r = 1. If f1 g2 − f2 g1 < 0, then similar arguments hold. We have N (r) > N (0) for any r > 0 and N (−1) is a minimum. The following corollary immediately follows from Lemma 4.1. Corollary 4.2. Let F and G be nonzero complex univariate polynomials in t, and α be a nonzero complex number. Put ζ = α/|α|. (That is, |ζ| = 1.) Then, the following inequality holds. 2 8 (t − ζ)F tF (t − α)F ≥ min , . (t − α)G (t − ζ)G tG When polynomials F and G are real, the following lemma holds. Lemma 4.3. Let F and G be nonzero real univariate polynomials in t, and r be a ﬁxed nonzero real number. Deﬁne ν : C → R by (t − rζ)F ν(ζ) := . (t − rζ)G We consider ν(ζ) a function on the unit circle in C. Then, ν(ζ) has a minimum at ζ = −1 or 1. Proof. Since ν(ζ) is nonnegative, it is suﬃcient to prove that N (ζ) = ν(ζ)2 has a minimum at ζ = −1 or 1. From Equation (4.5) and considering F , G ∈ R[t], we have (t − rζ)F 2

=

f1 (ζ + ζ) + f2 ,

(t − rζ)G

=

g1 (ζ + ζ) + g2 ,

2

where f2 = (r + 1)F , g2 = (r + 1)G2 and f1 , g1 ∈ R. Put s = ζ + ζ (∈ R). ˜ (s), which is a function of s (−2 ≤ s ≤ 2). Then we have We can write N (ζ) as N ˜ (s) = f1 g2 − f2 g1 , N (g1 s + g2 )2 ˜ (s) is monotonic on [−2, 2]. Therefore, it has a minimum at s = −2 or That is, N 2, which corresponds to ζ = −1 or 1, respectively. 2

2

2

Combining Lemmas 4.1 and 4.3, we can easily see that the following corollary holds. Corollary 4.4. Let F and G be nonzero real univariate polynomials, and α be a complex number. Then, the following inequality holds. 2 8 (t − α)F (t − 1)F (t + 1)F ≥ min , . (t − α)G (t − 1)G (t + 1)G Finally, we describe the following algorithm.

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S.M. Rump and H. Sekigawa

Algorithm 4.5. Given a real polynomial P (t) = (t − α)(t − α)P0 (t), where α ∈ C, ∈ R, this algorithm constructs F ∈ R[t] satisfying the following conditions. 1. The degree of F is two. 2. Both zeros of F lie on the unit circle. 3. F satisﬁes the following inequality. P Q F P0 Q ≥ . P F P0 Step 1.

Put ζ = α/|α|. If (t − ζ)(t − α)P0 Q (t − α)(t − α)P0 Q ≥ , (t − α)(t − α)P0 (t − ζ)(t − α)P0 then go to Step 2. Otherwise, go to Step 3.

Step 2. Step 2.1. If (t − ζ)(t − α)P0 Q (t − ζ)(t − ζ)P0 Q ≥ , (t − ζ)(t − α)P0 (t − ζ)(t − ζ)P0 then terminate with the output (t − ζ)(t − ζ). Otherwise, go to Step 2.2. Step 2.2. If t(t − 1)P0 Q t(t − ζ)P0 Q ≥ , t(t − ζ)P0 t(t − 1)P0 then put b1 = 1. Otherwise put b1 = −1. If t(t − b1 )P0 Q (t − 1)(t − b1 )P0 Q ≥ , t(t − b1 )P0 (t − 1)(t − b1 )P0 then put b2 = 1. Otherwise put b2 = −1. Terminate with the output (t − b1 )(t − b2 ). Step 3.

If

t(t − α)P0 Q t(t − 1)P0 Q ≥ , t(t − α)P0 t(t − 1)P0 then put b3 = 1. Otherwise, put b3 = −1. If t(t − b3 )P0 Q (t − 1)(t − b3 )P0 Q ≥ , t(t − b3 )P0 (t − 1)(t − b3 )P0 then put b4 = 1. Otherwise put b4 = −1. Terminate with the output (t − b3 )(t − b4 ).

The validity of the algorithm is as follows. In Step 2.1, if the inequality does not hold, then we have t(t − ζ)P0 Q (t − ζ)(t − α)P0 Q ≥ (t − ζ)(t − α)P0 t(t − ζ)P0 from Corollary 4.2.

The Toeplitz and the Unstructured Condition Number

411

In Step 2.2, the following inequalities hold from Corollary 4.4. t(t − b1 )P0 Q (t − b1 )(t − b2 )P0 Q t(t − ζ)P0 Q ≥ ≥ . t(t − ζ)P0 t(t − b1 )P0 (t − b1 )(t − b2 )P0 In Step 3, the inequality t(t − α)P0 Q (t − α)(t − α)P0 Q ≥ (t − α)(t − α)P0 t(t − α)P0 holds from Corollary 4.2. Furthermore, the inequalities t(t − α)P0 Q t(t − b3 )P0 Q (t − b3 )(t − b4 )P0 Q ≥ ≥ t(t − α)P0 t(t − b3 )P0 (t − b3 )(t − b4 )P0 hold from Corollary 4.4. Proof of Theorem 2.7. It is suﬃcient to show that the following two statements hold. 1. Given P = (t − a)P0 , where a ∈ R, we can construct a real polynomial R = (t − b)P0 (b = 1 or −1) satisfying the following inequality. P Q RQ ≥ . P R 2. Given P = (t − α)(t − α)P0 , where α ∈ C, ∈ R, we can construct a real polynomial R = F P0 with the inequality P Q RQ ≥ , P R where F is a univariate real polynomial of degree two with both zeros on the unit circle. The ﬁrst statement and the second statement follow from Corollary 4.4 and Algorithm 4.5, respectively. To prove Proposition 2.12, we need some lemmas. Lemma 4.6. Let P (t) be a real univariate polynomial of degree n. For an integer m > n, the equality m 1 |P (ωζ k )|2 P 2 = m k=1

holds, where ω ∈ C lies on the unit circle and ζ is a primitive mth root of unity. Lemma 4.7. For arbitrary > 0, there exists θ > 0 such that the inequality 1 − 2(1 − )x ≥ holds for θ ≥ x ≥ 0.

1 − sin x 1 + sin x

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S.M. Rump and H. Sekigawa

Proof. Since

2 sin x 1 − sin x =1− , 1 + sin x 1 + sin x the inequality is equivalent to sin x ≥ (1 − )x. 1 + sin x

(4.6)

As x tends to 0, sin x 1 → 1, → 1, x 1 + sin x hold. Therefore, for given > 0, there exists θ > 0 such that the inequality (4.6) holds for θ ≥ x ≥ 0. Lemma 4.8. For arbitrary > 0, there exists θ > 0 such that the inequalities exp(−x) ≥ 1 − x ≥ exp(−(1 + )x) hold for θ ≥ x ≥ 0. Lemma 4.9 (Jordan’s Inequality). For π/2 ≥ x ≥ 0, x ≥ sin x ≥

2x . π

Proof of Proposition 2.12. First we prove the proposition when n is odd. It is suﬃcient to show that for any > 0, there exists an integer N such that the inequalities

√ n2/3 2 1 1 n √ exp − + √ − 2 n 2 1− 2 n √ (4.7) √ ˆ n 2 n n exp −(1 + )πn1/3 Q 1 1 √ − > > √ − ˆ n 2 n 2(1 + )π 2 1+ Q D hold for any odd integer √ n ≥ N. Let ζ be exp(π −1/n). Then we have (n−1)/2 2n ˆ n (1)2 1 ˆ k 2 1 ˆ k 2 Q 2 ˆ + Qn = |Q(ζ )| = |Qn (ζ )| . 2n 2n n k=1

k=1

ˆ n (ζ k−1 )|2 is as follows. ˆ n (ζ k )|2 and |Q The relation between |Q 1 1 1 12 n+1 12 n−1 1 k−1 1 1 + ζ− 2 1 1ζ 11 + ζ − 2 −k 1 k 2 k−1 2 k−1 2 ˆ ˆ ˆ |Qn (ζ )| = |Qn (ζ )| 1 )| 1 1 = |Qn (ζ 12 . n−1 12 n+1 1 k−1 1 1 +ζ 2 1 1ζ 11 + ζ 2 −k 1 Since the equalities |1 + ζ | = (1 + ζ )(1 + ζ j 2

j

−j

jπ ) = 2 1 + cos n

The Toeplitz and the Unstructured Condition Number hold for j ∈ N, we have

1 + cos

π 2

+

(2k−1)π 2n

413

(2k−1)π

ˆ n (ζ k−1 )|2 1 − sin 2n . = |Q (2k−1)π π 1 + sin (2k−1)π 1 + cos 2 − 2n 2n (4.8) First we show the upper bound. Take any > 0. Then, there exists an integer L such that the above lemma holds for θ = L−1/3 π. Take any n ≥ L. Since we have ˆ n (ζ k−1 )|2 ˆ n (ζ k )|2 = |Q |Q

π (2n2/3 − 1)π π (2k − 1)π ≥ 1/3 ≥ ≥ 1/3 2n 2n L n for n2/3 ≥ k ≥ 1, the following inequalities follow from Lemmas 4.7 and 4.8.

1 − sin (2k−1)π (1 − )(2k − 1)π (1 − )(2k − 1)π 2n ≤ 1 − ≤ exp − . n n 1 + sin (2k−1)π 2n

Therefore, for n2/3 ≥ k ≥ 1 we have

(1 − )(2k − 1)π k 2 k−1 2 ˆ ˆ )| exp − |Qn (ζ )| ≤ |Qn (ζ n

k (1 − )(2j − 1)π 2 ˆ exp − ≤ Qn (1) n j=1 ⎛ ⎞

k ˆ n(1)2 exp − (1 − )π k 2 . ˆ n (1)2 exp ⎝− (1 − )π (2j − 1)⎠ = Q =Q n n j=1 Since the inequality 1 − sin (2k−1)π 2n 1 + sin

(2k−1)π 2n

≤ 1 − sin

(2k − 1)π 2n

holds for (n−1)/2 ≥ k > n2/3 , the following inequalities follow from Lemmas 4.8 and 4.9.

1 − sin (2k−1)π 2k − 1 (2k − 1)π 2n ≤ exp − . ≤ exp − sin 2n n 1 + sin (2k−1)π 2n Hence, for (n − 1)/2 ≥ k > n2/3 , we have

k 2k − 1 2j − 1 k 2 k−1 2 2 ˆ ˆ ˆ )| exp − exp − ≤ Qn (1) |Qn (ζ )| ≤ |Qn (ζ n n j=1 ⎛ ⎞ 2 k ˆ n (1)2 exp − k . ˆ n (1)2 exp ⎝− 1 (2j − 1)⎠ = Q = Q n j=1 n

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S.M. Rump and H. Sekigawa

Therefore, the following inequality holds.

ˆ n (1)2 ˆ n (1)2 n (1 − )π 2 Q Q exp − + k 2n n n 2/3

k=1

ˆ n (1) Q n

(n−1)/2

2

+

k=n2/3 +1

2 k ˆ n 2 . ≥ Q exp − n

Here, n2/3

k=1

(1 − )π 2 < k exp − n

holds since

4

∞

0

@

n 1 (1 − )π 2 x dx = exp − n 2 1−

@ 4 ∞ 1 1 π 2 exp(−cx )dx = √ exp(−x )dx = c 0 2 c 0 holds for c > 0. Then we have 2 2 2 4 ∞ 4 ∞ (n−1)/2 k x x exp − exp − x exp − < dx < dx n n n 2/3 2/3 n n k=n2/3 +1 2 ∞

x n2/3 2 n n . = − exp − = − exp − 2 n 2 n n2/3 4

∞

2

Hence, the following inequality holds.

2/3 2 1 1 1 n 2 ˆ ˆ n 2 . Q(1) − exp − + 5 > Q 2n 2 (1 − )n 2 n Therefore, we obtain the upper bound. That is, the inequality

√ ˆ n 2 √n 1 1 n2/3 2 Q n √ + √ − exp − > ˆ n (1)2 2 n 2 1− 2 n Q

(4.9)

holds for n ≥ L. Next, we show the lower bound. From (4.8) we have

2 j (2k − 1)π ˆ n (ζ j )|2 > Q ˆ n (1)2 1 − sin |Q . 2n k=1

Take any > 0. Then, there exists an integer M such that the above lemma holds for θ = M −1/3 π. Take any n ≥ M . Since for n2/3 ≥ k ≥ 1 we have π (2k − 1)π π (2n2/3 − 1)π ≥ , ≥ ≥ 2n 2n M 1/3 n1/3 the following inequalities follow from Lemma 4.8.

(2k − 1)π (2k − 1)π −(1 + )(2k − 1)π 1 − sin ≥1− ≥ exp . 2n 2n 2n

The Toeplitz and the Unstructured Condition Number

415

Hence, for n2/3 ≥ k ≥ 1 we have ⎞ ⎛

k 2 −(1 + )(2k − 1)π ˆ n (ζ k )|2 > Q ˆ n (1)2 exp −(1 + )πj ˆ n (1)2 exp ⎝ ⎠=Q |Q . n n j=1 Therefore, the following inequalities hold.

n n 1 ˆ k 2 1 −(1 + )πk 2 2 ˆ |Qn (ζ )| > Qn (1) exp n n n 2/3

ˆ n 2 Q

>

2/3

k=1

=

k=1

2/3 2 n

ˆ n (1) Q n

exp

k=1

−(1 + )πk2 n

.

The following estimation holds.

n2/3

exp

k=1

−(1 + )πk 2 n

4

n2/3 +1

exp

> 1

−(1 + )πx2 n

dx.

For a > 0 and c ≥ 1 we have 4 ∞ 4 1 4 ∞ 4 a exp(−cx2 )dx = exp(−cx2 )dx − exp(−cx2 )dx − exp(−cx2 )dx 1 0 0 a √ 4 ∞ π √ −1− > x exp(−cx2 )dx 2 c a and

∞

exp(−cx2 ) exp(−ca2 ) . x exp(−cx2 )dx = − = 2c 2c a a Hence, the inequalities (1+)π

√ 2/3 2 n2/3 n exp − 2 (n + 1) n −(1 + )πk n √ −1− > exp n 2(1 + )π 2 1 + k=1 √ n exp −(1 + )πn1/3 n √ > −1− 2(1 + )π 2 1+ 4

∞

hold. Therefore, we have ˆ n 2 Q

>

> >

2/3

n2/3 ˆ n (1)2 n 1 ˆ Q −(1 + )πj 2 k 2 |Qn (ζ )| > exp n n n k=1 k=1 2/3

4 ˆ n (1)2 n +1 −(1 + )πx2 Q exp dx n n 1 1/3 exp −(1 + )πn 1 1 2 ˆ n (1) 5 . − − Q 2(1 + )π 2 n(1 + ) n

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S.M. Rump and H. Sekigawa

Then, we obtain the lower bound. That is, the inequality √ ˆ n 2 √n n exp −(1 + )πn1/3 1 1 Q −√ − > √ ˆ n (1)2 2(1 + )π n 2 1+ Q

(4.10)

holds for n ≥ M . Combining (4.9) and (4.10), we have the statement (4.7) for N = max{L, M } when n is odd. √ Next we prove the statement when n is even. Let ζ4n be exp(π −1/2n). Note 2 that ζ4n is a primitive 2nth root of unity. Then, we have 1 ˆ 2k−1 2 1 ˆ 2k−1 2 |Q(ζ4n )| = |Qn (ζ4n )| . 2n n n/2

2n

ˆ n 2 = Q

k=1

The relation between

ˆ n (ζ 2k−1 )|2 ˆ n (ζ 2k+1 )|2 = |Q |Q 4n 4n Since

k=1

ˆ n (ζ 2k+1 )|2 |Q 4n

and

ˆ n (ζ 2k−1 )|2 |Q 4n

is as follows.

2k−1 −n−1 2 −n−2k 2 |ζ4n + ζ4n | | ˆ n (ζ 2k−1 )|2 |1 + ζ4n = |Q . 4n 2k−1 n−1 2 n−2k 2 |ζ4n + ζ4n | |1 + ζ4n |

jπ j 2 j −j | = (1 + ζ4n )(1 + ζ4n ) = 2 1 + cos , |1 + ζ4n 2n

we have π ˆ n (ζ 2k+1 )|2 = |Q ˆ n (ζ 2k−1 )|2 1 + cos( 2 + |Q 4n 4n 1 + cos( π2 −

kπ ) n kπ ) n

ˆ n (ζ 2k−1 )|2 1 − sin n . = |Q 4n 1 + sin kπ n kπ

From similar arguments for odd n, given > 0 there exists an integer N such that the following inequalities hold for any even integer n ≥ N .

√ ˆ n 2 √ n n n2/3 2 Q 1 √ exp − > − ˆ n (ζ4n )|2 2 n 2 1− |Q √ n exp −(1 + )πn1/3 1 1 . > √ −√ − n 2(1 + )π 2 1+ That is, we have

ˆ n 2 √n Q 1 → 2 ˆ 2 |Qn (ζ4n )|

(4.11)

as n tends to inﬁnity. According to the following Lemma, we have ˆ n (ζ4n )|2 |Q = 1, ˆ n 2 n→∞ Q lim

D

and combining with (4.11), we have the statement. Lemma 4.10.

ˆ n (ζ4n )|2 |Q = 1. ˆ n 2 n→∞ Q lim

D

The Toeplitz and the Unstructured Condition Number

417

Proof. Since the following inequalities

n/2

ˆ n (1) = Q

n/2 2k−1 (1 + ζ4n ),

ˆ n (ζ4n ) = Q

k=−n/2+1

2k−1 (ζ4n + ζ4n )

k=−n/2+1

hold, we have

n/2

ˆ n (1)2 = Q

n/2 2k−1 2 (1 + ζ4n ) =

2k−1 2 −2k+1 2 (1 + ζ4n ) (1 + ζ4n )

k=1

k=−n/2+1

2 n/2 (2k − 1)π = , 2 + 2 cos 2n k=1

n/2

ˆ n (ζ4n )|2 = |Q

n/2 2k−2 2 |1 + ζ4n | =

2k−2 2 −2k 2 |1 + ζ4n | · |1 + ζ4n |

k=1

k=−n/2+1

2 π (2k − 1)π 2 cos + 2 cos = . 2n 2n n/2

k=1

Therefore, the following inequalities hold. 2 2 2 n/2 cos π + cos (2k−1)π n/2 1 − π + cos (2k−1)π ˆ n (ζ4n )|2 2n 2n 8n2 2n |Q = ≥ 2 2 ˆ n (1)2 Q k=1 k=1 1 + cos (2k−1)π 1 + cos (2k−1)π 2n 2n ⎛ ⎞2

n n/2 ⎜ π2 π2 ⎟ = > 1 − . ⎝1 − ⎠ 2 8n2 k=1 8n2 1 + cos (2k−1)π 2n That is, we have 1≥

ˆ n(ζ4n )|2 |Q > ˆ n (1)2 Q

n π2 1− 2 . 8n

Therefore, we have ˆ n (ζ4n )|2 |Q →1 ˆ n (1)2 Q as n tends to inﬁnity.

Acknowledgement The ﬁrst author wishes to thank Prashant Batra, Ludwig Elsner and Arnold Sch¨ onhage for their valuable hints and fruitful discussion. The authors thank Kyoko Makino for computing the rigorous lower and upper bounds in Table 1. Furthermore our thanks to two unknown referees for their constructive comments.

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References [1] M. Berz. From Taylor series to Taylor models. In Nonlinear problems in accelerator physics, AIP Conference proceedings, number CP405, pages 1–27, 1997. [2] A. B¨ ottcher and S. Grudsky. Spectral properties of banded Toeplitz matrices. SIAM, Philadelphia, 2005. [3] A. B¨ ottcher and S. Grudsky. Structured condition numbers of large Toeplitz matrices are rarely better than usual condition numbers. Numerical Linear Algebra and its Applications, 12:95–102, 2005. [4] A. B¨ ottcher and K. Rost. Topics in the numerical linear algebra of Toeplitz and Hankel matrices. Mitt. Ges. Angew. Math. Mech., 27(2):174–188, 2004. [5] D.W. Boyd. Two sharp inequalities for the norm of a factor of a polynomial. Mathematika, 39:341–349, 1992. [6] D.W. Boyd. Sharp inequalities for the product of polynomials. Bull. London Math. Soc., 26:449–454, 1994. [7] S. Dallwig, A. Neumaier, and H. Schichl. GLOPT – a program for constrained global optimization. In I.M. Bomze et al., editor, Developments in global optimization, pages 19–36. Kluwer Academic Publishers, 1997. [8] I. Gohberg and A.A. Semencul. The inversion of ﬁnite Toeplitz matrices and their continual analogues. Matem. Issled, 7:201–223, 1972. [9] G. Heinig. private communication. [10] D.J. Higham and N.J. Higham. Backward error and condition of structured linear systems. SIAM J. Matrix Anal. Appl., 13(1):162–175, 1992. [11] E. Kaltofen, L. Bin, Y. Zhengfeng, and Z. Lihong. Exact certiﬁcation in global polynomial optimization via sums-of-squares of rational functions with rational coeﬃcients. submitted for publication, 2009. [12] H. Kneser. Das Maximum des Produkts zweier Polynome. In Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl., pages 426–431, 1934. [13] A. Neumaier. Complete search in continuous global optimization and constraint satisfaction. Acta Numerica, 13:271–369, 2004. [14] S.M. Rump. Structured perturbations Part I: Normwise distances. SIAM J. Matrix Anal. Appl. (SIMAX), 25(1):1–30, 2003. [15] S.M. Rump. A Model Problem for Global optimization. submitted for publication, 2009. [16] E.B. Saﬀ and T. Sheil-Small. Coeﬃcient and integral mean estimates for algebraic and trigonometric polynomials with restricted zeros. J. London Math. Soc., 9:16–22, 1974. [17] T. Sheil-Small. Complex polynomials, volume 75 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2002. Siegfried M. Rump Institute for Reliable Computing Hamburg University of Technology Schwarzenbergstraße 95 D-21071 Hamburg, Germany

The Toeplitz and the Unstructured Condition Number and Waseda University Faculty of Science and Engineering 3–4–1 Okubo, Shinjuku-ku Tokyo 169–8555, Japan e-mail: [email protected] Hiroshi Sekigawa NTT Communication Science Laboratories Nippon Telegraph and Telephone Corporation 3–1 Morinosato-Wakamiya, Atsugi-shi Kanagawa, 243-0198, Japan e-mail: [email protected]

419

Operator Theory: Advances and Applications, Vol. 199, 421–430 c 2010 Birkh¨ auser Verlag Basel/Switzerland

A New Algorithm for Finding Positive Eigenvectors for a Class of Nonlinear Operators Associated with M-matrices Yuriy V. Shlapak Abstract. In this paper we state the suﬃcient conditions for the existence and uniqueness of positive eigenvectors for a class of nonlinear operators associated with M-matrices. We also show how to construct a convergent iterative process for ﬁnding these eigenvectors. The details of numerical implementation of this algorithm for some spectral methods of discretization of elliptic partial diﬀerential equations are also discussed. Some results of numerical experiments for the Gross-Pitaevskii Equation with non-separable potentials in a rectangular domain are given in the end of the paper. Mathematics Subject Classiﬁcation (2000). 47J10. Keywords. Positive eigenvectors, Nonlinear operators, M-matrices, Monotone ﬁxed point theorem, Gross-Pitaevskii equation.

1. Introduction Many problems of the modern physics require ﬁnding positive eigenvectors of some nonlinear elliptic operators. After the discretization these problems can be reduced to problems of ﬁnding positive eigenvectors of nonlinear operators acting from Rn to Rn . We consider the following problem: ﬁnd a column vector X ∈ Rn that satisﬁes the equation AX + V ◦ X + F (X) = λX (1.1) where λ is a real positive constant (eigenvalue), A is an M-matrix of size n, symbol ◦ is used to denote the Hadamard product of two matrices, V = [v1 , . . . , vn ]T is a column vector that is called a potential, and F (X) is a nonlinear vector function from Rn to Rn that depends only on the vector X. In addition to ﬁnding the vector X, we would like to establish conditions that are suﬃcient for the existence and uniqueness of a vector X in this problem.

422

Y.V. Shlapak

To prove the main result of this paper, we use techniques that are similar to ones given in [1]. We consider a more complex equation (1.1), which has a linear term V ◦ X in it. This is a generalization of the equation considered in [1]. The discretizations of many important equations of physics have a linear term in them. For example, the Gross-Pitaevskii Equation from quantum physics −Δu + V (x, y, z)u + ku3 = λu

(1.2)

in one-dimensional case can be discretized by using the 3-point central ﬁnite diﬀerences and a uniform mesh with the step h into the equation (1.1) with the matrix A of the form: ⎤ ⎡ 2 −1 0 ⎥ ⎢ −1 2 −1 ⎥ ⎢ ⎥ ⎢ . . 1 .. .. ⎥ −1 A = 2 ·⎢ ⎥ ⎢ h ⎢ ⎥ . .. ⎣ 2 −1 ⎦ 0 −1 2 We should point out that the algorithm described in [1] and [2] allows practical computation of positive eigenvectors for some special types of equation (1.1). In this paper we provide both the theoretical justiﬁcation and practical algorithm for ﬁnding positive eigenvectors of equation (1.1) in its most general form. We can reformulate the problem of ﬁnding a solution of equation (1.1) as a problem of ﬁnding a ﬁxed point of some transformation, namely we need to ﬁnd X ∈ Rn such that X = S(X). We deﬁne S(X) by the formula S(X) = (cI + A)−1 ((c + λ)X − V ◦ X − F (X))

(1.3)

where c > 0 is some positive constant (usually we will choose it to be a large number). In order to prove the main theorem of this article, we will also need a Monotone Fixed Point Theorem [3] applied in the context of our problem. Theorem 1.1 (Monotone Fixed Point Theorem). Consider a space Rn with the partial order relation < deﬁned in the following way: for any two vectors X = [x1 , . . . , xn ]T ∈ Rn and Y = [y1 , . . . , yn ]T ∈ Rn we will say that X is smaller than Y (denoted as X < Y ) if xi ≤ yi for all i = 1, . . . , n and xi < yi for at least one i. If X ∈ Rn , Y ∈ Rn and X < Y we can deﬁne the interval [X, Y ] ⊂ Rn in the following way: we will say that T ∈ [X, Y ] if X ≤ T ≤ Y . Let Y ∈ Rn and Z ∈ Rn are such that Y < Z and let S: Rn → Rn be deﬁned and continuous on the interval [Y, Z] and suppose the following properties are satisﬁed 1) Y < S(Y ) < Z 2) Y < S(Z) < Z 3) Y ≤ X1 < X2 ≤ Z implies Y < S(X1 ) < S(X2 ) < Z Then a) the ﬁxed point iteration Xk = S(Xk−1 ) with X0 = Y converges: Xk → X∗ , S(X∗ ) = X∗ , Y < X∗ < Z

A New Algorithm for Finding Positive Eigenvectors

423

b) the ﬁxed point iteration Xk = S(Xk−1 ) with X0 = Z converges: Xk → X ∗ , S(X ∗ ) = X ∗ , Y < X ∗ < Z c) if X is a ﬁxed point of S in [Y, Z] then X∗ ≤ X ≤ X ∗ d) S has a unique ﬁxed point in [Y, Z] if and only if X∗ = X ∗

2. Monotone iteration for ﬁnding positive eigenvector Now we are ready to prove our main result of this article. It will state the existence and uniqueness of the positive solution of equation (1.1) and the convergence of the iteration sequence Xn+1 = S(Xn ), where S(X) is deﬁned by (1.3), to this solution. As was mentioned above, this theorem is a generalization of a theorem from [1], which is a particular case of our theorem when V = 0. Theorem 2.1 (Monotone iteration for ﬁnding positive eigenvector). Suppose in the equation (1.1) A is an M-matrix, μ is the smallest positive eigenvalue of the matrix A. Now we will use the notation X = [x1 , . . . , xn ]T and V = [v1 , . . . , vn ]T . Let λ > μ + max vi , and let 1≤i≤n

⎡ ⎢ ⎢ F (X) = ⎢ ⎣

f1 (x1 ) f2 (x2 ) .. .

⎤ ⎥ ⎥ ⎥ ⎦

(2.1)

fn (xn ) be such a vector function that for i = 1, . . . , n the components fi : (0, ∞) → (0, ∞) are C 1 -functions satisfying conditions fi (t) fi (t) = 0, lim =∞ t→∞ t→0 t t lim

(2.2)

Then (1.1) has a positive solution. If, in addition to the conditions given above, for i = 1, . . . , n we have fi (s) fi (t) < whenever 0 < s < t (2.3) s t then a positive solution of (1.1) is unique and there exists a vector X0 and a positive constant c such that the sequence Xn+1 = S(Xn ), where S(X) = (cI + A)−1 ((c + λ)X − V ◦ X − F (X)), converges to the unique positive solution of (1.1). Proof. The proof is based on showing that S(X) satisﬁes the conditions of the Monotone Fixed Point Theorem (Theorem 1.1). It will guarantee the existence of the positive solution and its uniqueness. First of all, we choose a real number β1 small enough so that (λ − vi − μ)(β1 pi ) > fi (β1 pi ) for all i = 1, . . . , n and we choose β2 > β1 large enough so

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Y.V. Shlapak

that (λ − vi − μ)(β2 pi ) < fi (β2 pi ) for all i = 1, . . . , n. This is always possible in view of limits (2.2). We also choose a positive number c such that c > max

1≤i≤n

sup β1 pi ≤t≤β2 pi

|fi (t)|

− λ + max vi 1≤i≤n

(2.4)

and then reformulate the problem of ﬁnding a solution of equation (1.1) into a problem of ﬁnding a ﬁxed point of the transformation S : Rn → Rn , where S(X) is deﬁned as in (1.3), i.e., S(X) = (cI + A)−1 ((c + λ)X − V ◦ X − F (X))

(2.5)

To prove the existence of a ﬁxed point of (1.3) it is enough to show that S(X) satisﬁes the conditions of Monotone Fixed Point Theorem given above with Y = β1 p and Z = β2 p. First of all, let us show that the condition 1) of Monotone Fixed Point Theorem is satisﬁed, i.e., Y < S(Y ) < Z. In the proof below we will use the obvious fact that p = (A + cI)−1 (c + μ)p. Since (A + cI)−1 u > 0 whenever u > 0 (i.e., (A + cI)−1 is a positive matrix) it suﬃces to show that (c + μ)(β1 p) < (c + λ)(β1 p) − V ◦ (β1 p) − F (β1 p) < (c + μ)(β2 p)

(2.6)

We start from proving the left part of this double inequality. The argument below is valid for any i = 1, . . . , n. From (λ − vi − μ)(β1 pi ) > fi (β1 pi ) we have (λ − vi − μ)(β1 pi ) − fi (β1 pi ) > 0 and so (c + μ)(β1 pi ) + (λ − vi − μ)(β1 pi ) − fi (β1 pi ) > (c + μ)(β1 pi ). After cancelling out the term μβ1 pi in the left part we get (c + λ)(β1 pi ) − vi (β1 pi ) − fi (β1 pi ) > (c + μ)(β1 pi ) which is exactly the componentwise notation of the left part of the double inequality (2.6). Now we will prove the right part of the double inequality (2.6). By our choice of β2 for any i = 1, . . . , n we have fi (β2 pi ) > (λ − vi − μ)(β2 pi ). It can be rewritten as fi (β1 pi ) + (fi (β2 pi ) − fi (β1 pi )) > (λ − vi − μ)(β2 pi ) or, if we estimate the change in f by its derivative multiplied by the change in argument of f , and use (2.4), we will obtain that fi (β1 pi ) + (c + λ − vi )((β2 pi ) − (β1 pi )) > (λ − vi − μ)(β2 pi ) and after moving some terms into the left part we get fi (β1 pi ) − (c + λ − vi )(β1 pi ) > (λ − vi − μ)(β2 pi ) − (c + λ − vi )(β2 pi ). After simpliﬁcation it becomes fi (β1 pi )−(c+λ−vi )(β1 pi ) > −(μ+c)(β2 pi ) or, if we multiply it by −1, it becomes (μ + c)(β2 pi ) > (c + λ)(β1 pi ) − vi (β1 pi ) − fi (β1 pi ), which is exactly the right part of equality (2.6). We also need to prove that the second condition of Monotone Fixed Point Theorem is satisﬁed, namely, Y < S(Z) < Z. Due to the fact that (A + cI)−1 is a positive matrix it suﬃces to show only that (c + μ)(β1 p) < (c + λ)(β2 p) − V ◦ (β2 p) − F (β2 p) < (c + μ)(β2 p)

(2.7)

Let us show now that the right part of this double inequality holds. The argument below is valid for any i = 1, . . . , n. From (λ − vi − μ)(β2 pi ) < fi (β2 pi ) we have (λ − vi − μ)(β2 pi ) − fi (β2 pi ) < 0 and so we can write (c + μ)(β2 pi ) + (λ − vi −

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425

μ)(β2 pi ) − fi (β2 pi ) < (c + μ)(β2 pi ) and then after cancelling out the term μβ2 pi in the left part we get (c + λ)(β2 pi ) − vi (β2 pi ) − fi (β2 pi ) < (c + μ)(β2 pi ), which is exactly the right part of the double inequality (2.7). Now we will prove that the left part of the double inequality (2.7) holds. From our choice of β1 we have fi (β1 pi ) < (λ − vi − μ)(β1 pi ), which can be written as fi (β2 pi ) + (fi (β1 pi ) − fi (β2 pi )) < (λ − vi − μ)(β1 pi ) or, if we estimate the change in f by its derivative multiplied by the change in argument of f , and use (2.4), we will get fi (β2 pi ) + (c + λ − vi )((β1 pi ) − (β2 pi )) < (λ − vi − μ)(β1 pi ). After moving some terms into the left part, it becomes fi (β2 pi ) − (c + λ − vi )(β2 pi ) < (λ − vi − μ)(β1 pi ) − (c + λ − vi )(β1 pi ) and after some simpliﬁcation it becomes fi (β2 pi ) − (c + λ − vi )(β2 pi ) < −(μ + c)(β1 pi ). Finally, if we multiply it by −1 we will get (μ + c)(β1 pi ) < (c + λ)(β2 pi ) − vi (β2 pi ) − fi (β2 pi ) which is exactly the componentwise notation of the left part of double inequality (2.7). Now we have to show that if β1 p ≤ X1 < X2 ≤ β2 p then S(X1 ) < S(X2 ). It will guarantee that the condition 3) of Monotone Fixed Point Theorem is satisﬁed. If we denote ith components of vectors X1 and X2 as x1i and x2i correspondingly then we can write the ith component of S(X2 )−S(X1 ) as (A+cI)−1 ((c+λ−vi )x2i − fi (x2i )−(c+λ−vi )x1i +fi (x1i )). Due to the fact that (A+cI)−1 is a positive matrix it suﬃces to show only that (c+λ−vi )x2i −fi (x2i )−(c+λ−vi )x1i +fi (x1i ) > 0 for each i = 1, . . . , n.. It can be shown by using some simple algebraic transformations and estimating the change in f by the maximum of its derivative multiplied by the change in argument of f and then using (2.4). For each i we have (c + λ − vi )x2i − fi (x2i ) − (c + λ − vi )x1i + fi (x1i ) = (c + λ − vi )(x2i − x1i ) − (fi (x2i − fi (x1i )) > (c + λ − vi )(x2i − x1i ) − (c + λ − vi )(x2i − x1i ) = 0. So β1 p ≤ X1 < X2 ≤ β2 p implies S(X1 ) < S(X2 ). We have checked that the conditions 1)-3) of Monotone Fixed Point Theorem are satisﬁed. It guarantees that there exists at least one ﬁxed point of the transformation deﬁned by (1.3) that will also be a solution of the equation (1.1). The Monotone Fixed Point Theorem also implies that if we choose X0 = β1 p or X0 = β2 p then the sequence Xn+1 = S(Xn ) will converge to a ﬁxed point of the transformation S(X) (which will also be a solution of equation (1.1)). And ﬁnally, it guarantees that such ﬁxed point(s) of S(X) will lie between β1 p and β2 p, which guarantees the positivity of all components of the solution. Now we will prove the uniqueness of the positive solution under the conditions of Theorem 2.1. Suppose now that the condition (2.3), i.e., fi (t) fi (s) < whenever 0 < s < t s t is satisﬁed. We want to show that in this case for any two positive solutions X ∗ and X∗ it must be X ∗ = X∗ . Since both X ∗ and X∗ are solutions of the equation (1.1), we have AX ∗ + V ◦ X ∗ + F (X ∗ ) = λX ∗ AX∗ + V ◦ X∗ + F (X∗ ) = λX∗ .

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Pre-multiplying the ﬁrst equation by X∗T and the second equation by X ∗T and subtracting the second equation from the ﬁrst one we will get (using an obvious identity X ∗T ·V ◦ X∗ = X∗T ·V ◦ X ∗ ) that X∗T F (X ∗ ) = X ∗T F (X∗ ) or, equivalently, that using the componentwise notation we can write

n n fi (x∗i ) fi (xi∗ ) x∗i x∗i 0= (fi (x∗i )x∗i − fi (x∗i )x∗i ) = − x∗i xi∗ i=1 i=1 Since x∗i and xi∗ are always positive, the only case when this sum can be zero is when for each i = 1, . . . , n fi (xi∗ ) fi (x∗i ) = (2.8) x∗i xi∗ which, by condition (2.3), can only be true when X∗ = X ∗ . So the uniqueness of the positive solution of (1.1) under the condition (2.3) is also proven. The proof of Theorem 2.1 is completed.

3. The numerical implementation in two-dimensional case As an application of the algorithm described above, we consider solving a discretized boundary value problem for a nonlinear elliptic PDE in two dimensions. Suppose our domain is a square Ω = {(x, y)| − T ≤ x ≤ T, −T ≤ y ≤ T } and we need to ﬁnd a function u(x, y) such that for every (x, y) ∈ Ω it satisﬁes the equation −Δu(x, y) + V (x, y)u(x, y) + f (u(x, y)) = λu(x, y) (3.1) where Δ is Laplace operator, V (x, y) is a function that is called a potential, f is a nonlinear function and λ is a positive constant. The function u must also satisfy the following boundary condition: u|Γ = 0

(3.2)

where Γ is the boundary of a square Ω. We approximate the function u(x, y) by a linear combination of normalized Legendre polynomials: N

u(x, y) =

αij pi

x

i,j=1

T

pj

y T

,

(3.3)

where pi (x) is the normalized Legendre polynomial of degree i. We use the set of collocation points {(xk , ym )| k, m = 1, . . . , N + 1}, where xk and yk are zeros of the N + 1-st normalized Legendre polynomial pN +1 (x) that are multiplied by T. Naturally, we have u(xk , ym ) =

N i,j=1

αij pi

x k

T

pj

y m

T

(3.4)

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427

and now we deﬁne U to be the matrix of values of function u at the collocation points, i.e., Ukm = u(xk , ym ). We similarly deﬁne matrix V as the matrix of values of potential V at the collocation points, i.e., Vkm = V (xk , ym ). It is widely known (see [4] or the Appendix of [2]) that if u is a linear combination of normalized Legendre polynomials as given in (3.4), and U is the matrix of values of u at the collocation points given above, then the matrices of values of 2 2 partial derivatives ∂∂xu2 and ∂∂yu2 at the collocation points can be found correspondingly as DU and U D, where D is a known square matrix that depends only on N and T . The matrix D plays the role of the double diﬀerentiation operator for our functions at the collocation points. The boundary value problem (3.1)–(3.2) at the collocation points can be written as the following matrix equation: −DU − U D + V ◦ U + f (U ) = λU,

(3.5)

where ◦ is a symbol of Hadamard product of two matrices. The matrix D is not an M-matrix, but our numerical experiments showed that the inverse of the matrix cI + D was a positive matrix in all our experiments. So the statement and the results of the Theorem 2.1 are also valid for the matrix equations that involve D instead of A. The iteration sequence generated by transformation (1.3) for ﬁnding the approximate solution of the equation (3.5) is deﬁned by relation cUn+1 − DUn+1 − Un+1 D = cUn + λUn − V ◦ Un − f (Un )

(3.6)

In order to ﬁnd Un+1 (when Un is known) we use a technique described in [2], [4]. This technique makes extensive use of properties of orthogonal polynomials at certain points. The numerical implementation of this technique is relatively straightforward and gives excellent results in terms of computational speed and precision. To illustrate the feasibility and eﬀectiveness of the method described in this paper, we found positive solutions of the Gross-Pitaevskii Equation (1.2) with non-separable potentials (i.e., potentials that cannot be expressed as sums of two functions that depend only on one spacial variable each). This type of the GrossPitaevskii Equation is more diﬃcult to work with than the one with a separable potential. For example, the method described in [1], [2] cannot be used for such equations. In our experiments, the domain was a square [−10, 10] × [−10, 10]. It took about 31 seconds for our personal computer to do 1000 iterations on a 128 by 128 mesh of collocation points. Some graphs and tables related to our numerical experiments are given next.

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Figure 1. The solution for V (x, y) = 0.6x2 + 0.4y 2 + 7e−(x+2)

2

−y 2

,

k = 10,

λ = 10

Table 1. Convergence results for V (x, y) = 0.6x2 + 0.4y 2 + 7e−(x+2)

2

−y 2

,

k = 10,

λ = 10,

Number of iterations 2-norm of the residue 2 14.4357 5 4.4489 10 3.7863 20 7.3529 50 46.2945 100 14.0350 200 0.6203 500 1.6815 ·10−5 1000 2.4052 ·10−9 2000 2.4054 ·10−9 5000 2.4054 ·10−9

c = 100

A New Algorithm for Finding Positive Eigenvectors

Figure 2. The solution for 3 5 0.6(x − 2)2 + 0.8(y + 3)2 , V (x, y) = 0.1

Table 2. Convergence results for 3 5 0.6(x − 2)2 + 0.8(y + 3)2 , k = 10, V (x, y) = 0.1

k = 10,

λ = 10,

Number of iterations 2-norm of the residue 2 28.1881 5 9.8392 10 5.2898 20 6.2834 50 21.7926 100 83.8192 200 15.5093 500 0.0366 1000 1.0538 ·10−6 2000 5.2324 ·10−9 5000 5.2321 ·10−9

429

λ = 10

c = 200

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Y.V. Shlapak

We should point out that increasing c in (3.6) slows down the convergence of our algorithm. However, as the condition (2.4) in Theorem 2.1 indicates, increasing c may be necessary to guarantee the convergence of the algorithm and our numerical experiments conﬁrmed the necessity of this requirement.

References [1] Choi Y.S., Koltracht I., McKenna P.J. A generalization of the Perron-Frobenius theorem for non-linear perturbations of Stiltjes Matrices. Contemporary Mathematics, Volume 281, 2001, pp. 325–330 [2] Choi Y.S., Javanainen J., Koltracht I., Kostrun M., McKenna P. J., Savytska N., A fast algorithm for the solution of the time-independent Gross-Pitaevskii equation. Journal of Computational Physics, 190 (2003), pp. 1–21 [3] Kantorovich L.V., Vulikh B.Z., Pinsker A.G., Functional Analysis in Semi-ordered Spaces. (in Russian Language). Moscow, GosIzdat Technico-Teoreticheskoi Literatury, 1950. [4] Gottlieb D., Orszag S.A., Numerical Analysis of Spectral Methods: Theory and Applications. Philadelphia, Society for Industrial and Applied Mathematics, 1977. Yuriy V. Shlapak Department of Mathematics University of Connecticut 196 Auditorium Road, Unit 3009 Storrs, CT 06269-3009, USA e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 199, 431–439 c 2010 Birkh¨ auser Verlag Basel/Switzerland

Hankel Minors and Pade Approximations Eugene Tyrtyshnikov To the memory of Georg Heinig

Abstract. Algebraic Pade theory is presented in a complete, brief and clear way as a corollary of one property of a sequence of nonzero leading minors in a semi-inﬁnite Hankel matrix associated with a formal series. Mathematics Subject Classiﬁcation (2000). 15A12; 65F10; 65F15. Keywords. Pade approximants; Hankel matrices; Toeplitz matrices; rational approximations.

1. Introduction Algebraic theory of Pade approximations is in fact a theory of submatrices in a semi-inﬁnite Hankel matrix. Basic assertions of this theory in [1] are formulated in an elegant and simple way. However, one could be disappointed by some proofs looking too tangled and even somewhat misleading. At the same time, algebraic facts are not in the focus of the extensive literature on rational approximations (see [7, 2, 3]) devoted chieﬂy to analytical questions and applications. The purpose of this paper is a complete, brief and transparent presentation of the algebraic Pade theory as a corollary of a nice property of leading minors of a Hankel matrix, which is probably best exposed in the enlightening book by G. Heinig and K. Rost [5]. This property is related to a problem of rank-preserving augmentations [6] and was expounded in [10] as a base of the “method of jumps” for the inversion of Hankel matrices with some of leading minors equal to zero. Besides a certain development of the “method of jumps”, in this paper we contribute with new proofs for a systematic presentation of the known statements of the algebraic Pade theory. Supported by RFBR grants 08-01-00115, 09-01-91332, 09-01-112058, the State Contract Π940 and Priority Research Programme of Department of Mathematical Sciences of Russian Academy of Sciences.

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E. Tyrtyshnikov

2. Series and matrices Consider a formal series f (x) =

∞

a i xi .

i=0

We deﬁne a Pade approximation of type (m, n) for f (x) as a pair of polynomials u(x) =

m

ui xi ,

v(x) =

i=0

n

vi xi

i=0

such that f (x)v(x) − u(x) = O(xm+n+1 )

(2.1)

under an additional restriction v(0) = 1.

(2.2)

It follows immediately that f (x) −

u(x) = O(xm+n+1 ). v(x)

Condition (2.1) is equivalent to the system of linear equations n

ai−j vj = 0,

m + 1 ≤ i ≤ m + n,

j=0

or, in matrix notation, ⎡ am+1 ⎢ am+2 ⎢ ⎣ ... am+n

am am+1 ... am+n−1

Taking into account the equation ⎡ ... am ⎣ ... ... am+n−1 . . .

... ... ... ...

⎤⎡ ⎤ am−n+1 v0 ⎥ ⎢ v1 ⎥ am−n ⎥ ⎢ ⎥ = 0. ⎦ ⎣. . .⎦ ... vn am

v0 = 1, we obtain ⎤⎡ ⎤ ⎡ ⎤ am−n+1 v1 am+1 ⎦ ⎣. . .⎦ = − ⎣ . . . ⎦ . ... vn am+n am

To clarify things, take m = n = 3, then ⎤⎡ ⎤ ⎡ ⎤ ⎡ v1 a4 a3 a2 a1 ⎣a4 a3 a2 ⎦ ⎣v2 ⎦ = − ⎣a5 ⎦ . a5 a4 a3 v3 a6 Each entry in the coeﬃcient matrix of this system is deﬁned by the diﬀerence of the row and column indices – such matrices are called Toeplitz matrices. By taking the columns in the reverse order we obtain a matrix whose entries are deﬁned by

Hankel Minors and Pade Approximations

433

the sum of row and column indices – such matrices are called Hankel matrices. By reversion of columns the above system reduces to the following one: ⎤⎡ ⎤ ⎡ ⎤ ⎡ v3 a4 a1 a2 a3 ⎣a2 a3 a4 ⎦ ⎣v2 ⎦ = − ⎣a5 ⎦ . a3 a4 a5 v1 a6 In the general case we obtain a system of the form ⎤⎡ ⎤ ⎡ ⎤ ⎡ vn am+1 am−n+1 . . . am ⎦ ⎣. . .⎦ = − ⎣ . . . ⎦ , ⎣ ... ... ... am v1 am+n . . . am+n−1

(2.3)

or, in short notation, (2.4) Amn vn = −amn . The Hankel matrix Amn is composed of the coeﬃcients of the formal series f (x) and deﬁned by m and n as follows: am is located in its lower left corner and n is the matrix order. If i < 0, then ai = 0 by deﬁnition. The right-hand side vector amn is the last column of the extended rectangular Hankel matrix [Amn , amn ]. Thus, the existence of a Pade approximation of type (m, n) is equivalent to the consistency of the linear system (2.4). The latter means that amn ∈ imAmn , and, by the Kronecker–Capelli theorem, is equivalent to the condition rankAmn = rank[Amn , amn ].

(2.5)

3. Jumps over zero minors Given a semi-inﬁnite Hankel matrix A = [ai+j−1 ],

1 ≤ i, j < ∞,

let us consider its leading submatrices. Let Ak be a leading submatrix of order k, and let a sequence of natural numbers n1 < n2 < · · · determine the orders of those and only those of them which are nonsingular. The method of jumps suggested in [10] is a scheme of the transition from some compact representation (suggested in [4] for Toeplitz matrices and then generalized to many classes of structured matrices, cf. [5]) for A−1 nk to a similar-style representation . It can be thought of as a jump over all intermediate singular leading for A−1 nk+1 submatrices, which explains the name. General questions of the design of fast algorithms for Hankel and Toeplitz matrices are considered, for instance, in [8, 9]. Algebraic properties of Hankel matrices needed to perform the above-discussed jump are presented in [5] together with a structure of null-spaces of Hankel matrices. They are closely connected with inﬁnite rank-preserving augmentations studied in [6].

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E. Tyrtyshnikov

The method of jumps stems from the following observation. Let p = nk and q = nk+1 . Since Ap is nonsingular, the system ⎡ ⎤⎡ ⎤ ⎡ ⎤ a1 . . . ap s1 ap+1 ⎣ ... ... ... ⎦ ⎣. . .⎦ = ⎣ . . . ⎦ ap . . . a2p−1 sp a2p has a unique solution. In other words, the columns from the 1st to pth truncated up to p elements are linearly independent, and the column p+1 truncated in the same way can be written as their linear combination with the coeﬃcients s1 , . . . , sp . It may happen that the same coeﬃcients can be used for the column p + 1 to be a linear combination of the preceding columns when truncated up to p + 1 or even larger number of elements. Theorem 3.1. Let r(p) ≥ p be the minimal size for truncation, in which the column p + 1 is not in the linear span of preceding columns. Then nk+1 = r(nk ). Proof. Let p = nk and r ⎡ a1 ... ⎢ ... ... ⎢ ⎢ ap ... ⎢ ⎢ ... ... ⎢ ⎣ ar−1 . . . ar ...

= r(p). Then ap ... a2p−1 ... ar+p−2 ar+p−1

ap+1 ... a2p ... ar+p−1 ar+p

⎤ 0 ⎥ −s1 ⎢. . .⎥ ⎥ ⎢ ⎥ ⎥⎢ ... ⎥ ⎢ 0 ⎥ ⎥⎢ ⎥ = ⎢ ⎥, ⎥ ⎣−sp ⎦ ⎢. . .⎥ ⎥ ⎢ ⎥ ⎦ 1 ⎣0⎦ γ ⎤

⎡

⎤

⎡

γ = 0.

This implies the equation ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Ar ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

−s1 ... ... −sp 1

⎤

1 −s1 ... ... −sp 1

1 ... ... ... ... ... ... ... . . . −sp 1

⎡ ⎥ ⎥ ⎢ ⎥ ⎢ ... ⎥ ⎢ ⎢ 1⎥ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

a1 a2 a2 a3 ... ... ap ap+1 γ ap+1 ap+2 ... ... ... ... ... γ ... ... ... γ . . . . . . . . . ar ar+1

... ap . . . ap+1 ... ... . . . a2p−1 ... a2p ... ... ... ... . . . ar+p−1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎦

(3.1)

If p = nk then the matrix Ap is nonsingular. Therefore, its augmentation Ar is nonsingular if and only if γ = 0. Corollary 3.2. If nk ≤ n ≤ nk+1 then dim kerAn = min{n − nk , nk+1 − n}. Proof. We derive directly from (3.1) that An times a nonsingular matrix is a matrix having min{n − nk , nk+1 − n} zero columns and all nonzero columns producing a linearly independent system.

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435

Denote by Aˆn the augmentation of An of the following form: ⎡ ⎤ a1 . . . an an+1 ... ... ⎦. Aˆn = ⎣. . . . . . an . . . a2n−1 a2n Corollary 3.3. If nk ≤ n < nk+1 , then the equality of ranks rankAn = rankAˆn

(3.2)

takes place when and only when n − nk < nk+1 − n.

(3.3)

Proof. According to the Kronecker–Capelli theorem, the equality (3.2) is equivalent to the claim for the last column of the extended matrix Aˆn to be a linear combination of the preceding columns. Let p = nk and q = nk+1 . Let p ≤ n = p + i < q, and note that the column p + 1 of Aq−1 is a linear combination of the preceding columns with the coeﬃcients s1 , . . . , sp . The same is valid for the column p + 1 of all those of its leading submatrices that contain Ap+1 . If p + 2i < q, then this holds true for the matrix Ap+2i . Allowing for the Hankel structure of matrices, we easily deduce that the column p + 1 + i – (the last column of the extended matrix Aˆp+i ) is in the linear span of the preceding p columns (with the same coeﬃcients s1 , . . . , sp ). The inequality p + 2i < q is equivalently to n − p < q − n. It remains to prove that (3.2) implies (3.3). In accordance with (3.2) the last column of the extended matrix Aˆn is in the linear span of the columns of An . In this case, when passing from An to An+1 the rank may increase by at most 1. By contradiction, let us admit that n − p ≥ q − p. On the base of Corollary 3.2, rankAn = n − min{n − p, q − n} = 2n − q, rankAn+1 = n + 1 − min{n + 1 − p, q − n − 1} = 2n − q + 2. Hence, rankAn+1 = rankAn + 2, which is impossible since the rank cannot inﬂate greater than by 1.

Corollary 3.4. If nk ≤ n < nk+1 , then the inequality (3.3) is fulﬁlled when and only when rankAn+1 − rankAn ≤ 1.

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4. An identity for determinants Let A be a matrix of order n and Aij be its submatrix of order n − 1 obtained by ruling out the row i and column j. Let Aik;jl denote a submatrix of order n − 2 appeared from A be deleting a pair rows with the indices i and k and a pair of columns with the indices j and l. The next result is known as the Sylvester identity. Theorem 4.1. Let i < k and j < l. Then det A det Aik;jl = det Aij det Akl − det Ail det Akj . Proof. Without loss of generality we may assume that i = j = n−1 and k = l = n. Let B = Aik;jl . Then the matrix A is of the form ⎤ ⎡ B v q A = ⎣ u c d⎦ . p g h Suppose ﬁrst that B is nonsingular. By block Gaussian elimination of u and p with the pivot B we ﬁnd ⎤ ⎡ ⎤⎡ ⎤ ⎡ B v q B v q I 0 0 ⎣−uB −1 1 0⎦ ⎣ u c d ⎦ = ⎣ 0 c1 d1 ⎦ , p g h −pB −1 0 1 0 g1 h 1 where h1 = h − pB −1 q,

c1 = c − uB −1 v,

g1 = g − pB −1 v,

d1 = d − uB −1 q.

Consequently, det Aij det Akl − det Ail det Akj = (det B)2 (h1 c1 − g1 d1 ) = det B det A. If B is singular then the target identity is valid as soon as we replace B with any nonsingular block Bε . Since Bε can be chosen arbitrarily close to B, we complete the proof by transition to the limit.

5. Table of minors Proceed with the study of an inﬁnite Hankel matrix A = [ai+j ] composed of ∞ coeﬃcients of the formal series f (x) = ai xi (if i < 0 then ai = 0). Recall i=0

that Amn is its Hankel submatrix of order n with the entry am in the lower left corner. We are interested to examine a semi-inﬁnite matrix C = [cmn ] collecting the minors of A: cmn = det Amn , 0 ≤ m, n < ∞. Set cm0 = 1. Lemma 5.1. cm,n+1 cm,n−1 = cm+1,n cm−1,n − c2mn . Proof. This equality is nothing else than the Sylvester identity for determinants (Theorem 4.1) applied to the Hankel matrix Am,n+1 and its submatrices after expunging the ﬁrst and last rows and columns, the Hankel property being kept with this special choice.

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The table of minors C for the Hankel matrix A is sometimes referred to as C-table [1]. The basic property of the C-table consists in a special structure of zeroes (zero minors of A). Let us call a window any ﬁnite or inﬁnite submatrix made up from the entries of contiguous rows and columns. A window is called a square window if it corresponds to a ﬁnite square submatrix or an inﬁnite one with inﬁnitely many both rows and columns. All the entries adjacent to a given window will be called a frame – these entries belong to a wider window with the rows and columns augmenting the given window. Above all, we need to investigate zero windows and nonzero frames – in the ﬁrst case all the entries in the window are zeroes, in the second case all the entries in the frame diﬀer from zero. Here and throughout, assume that a0 = 0. Then c0n = 0 for n ≥ 1 (the determinants of Hankel triangular matrices with a nonzero along the anti-diagonal). Moreover, let us agree that cm0 = 1 whenever m ≥ 0. Theorem 5.2. Any zero entry in the table of minors C belongs to a square window with a nonzero frame. Proof. Assume that cm,n−1 = cm+1,n = 0 and cm,n+1 = cm+1,n = 0. From Lemma 5.1 we ﬁnd cmn = 0. As a corollary, submatrices of the form

∗ 0 0 ∗ , 0 ∗ ∗ 0 must be zero. Combining this with the inequalities cm0 = 0 and c0n = 0 we conclude that any zero entry of C belongs to a rectangular window with a nonzero frame. It remains to prove that this window is in fact a square one. Let cmn = 0 is an entry of the frame located at the upper left corner. Assume that cm+r,n+r = 0 is one more entry of the same window’s frame and prove that it lies at the lower right corner. If it were not so, then we would have two options: (1) cm+r,n+r−1 = 0, or (2) cm+r−1,n+r = 0. Option (1). Note that Amn and Am+r,n+r are nonsingular leading submatrices in the Hankel matrix Am+r,n+r and the intermediate leading submatrices Am+i,n+i for 0 < i < r are singular. According to the method of jumps (Theorem 3.1), the column n + 1 of Am+r,n+r without the last row is a linear combination of the preceding columns. In this case cm+1,n = 0, cm+r+1,n+r = 0. This validates application of the method of jumps to nonsingular Hankel matrices Am+1,n and Am+1+r,n+r , and so we come to the conclusion that the column n + 2 of Am+r,n+r without the last row is a linear combination of the preceding columns starting from the 2nd one. By subtraction of these linear combinations from the columns n + 1 and n + 2 we do not change the determinant of Am+r,n+r and acquire two columns with zeroes except for the last row’s entries. Consequently, cm+r,n+r = det Am+r,n+r = 0, which contradicts to the initial assumption. Thus, cm+r,n+r−1 = 0. Option (2). Observe that cm+r,n+r+1 = 0 (otherwise cm+r,n+r = 0 by Lemma 5.1). Thus, Am,n+1 are Am+r,n+r+1 are nonsingular leading submatrices with singular

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intermediate submatrices. In line with the method of jumps, the column n + 2 in Am+r,n+r+1 without the last row is a linear combination of the preceding columns. Hence, the column n + 2 of Am+r,n+r without the last row is a linear combination of the preceding columns. As previously, the same holds true for the column n + 1 of the same matrix. Again we come to contradiction with the nonsingularity of Am+r,n+r . Therefore, cm+r−1,n+r = 0. All in all, both options (1) and (2) lead to contradiction. It means that we have simultaneously cm+r,n+r = 0,

cm+r,n+r−1 = 0,

cm+r−1,n+r = 0.

This proves that cm+r,n+r is located at the lower right corner of the zero window’s frame. Since the upper left corner is occupied by cmn , this is a square window. The case when cm+r,n+r = 0 for all r > 0 is merely simpler: a recursive use of Lemma 5.1 makes it clear that we deal with an inﬁnite square window.

6. Pade theory Pade theory provides us with the necessary and suﬃcient condition for the existence of a Pade approximation of type (m, n) in the terms of zero structure in the table of minors associated with the formal series f (x). As we already know, a condition of this kind is the consistency of the linear system (2.4). Obviously, the existence is guaranteed whenever cmn = det Amn = 0. The main result for the case cmn = 0 is formulated as follows. Theorem 6.1. Let cmn = 0 belong to a zero window with a nonzero frame whose upper left and lower right corners keep ckl = 0 and ck+r,l+r = 0, respectively. Then, Pade approximation of type (m, n) exists if and only if k + l < m + n < k + l + r. Proof. Assume that cst and cs+p,t+p belong to the nonzero frame of the given zero window and for some h > 0 we have m = s + h and n = t + h. The matrix Ast is a nonsingular leading submatrix in the nonsingular Hankel matrix As+p,t+p , and the submatrices As+i,t+i are all singular for 0 < i < p. From the method of jumps (see Corollary 3.3) it emanates that the compatibility condition (2.5) is equivalent to the inequality (t + h) − t < (t + p) − (t + h)

⇔

h < p/2.

It is fulﬁlled if and only if m + n < k + l + r.

Theorem 6.2. Let ckl and ck+r,l+r be the corner entries of a nonzero frame of some zero window of the table of minors, and assume that m ≥ k and n ≥ l satisfy the inequalities k + l ≤ m + n < k + l + r. Then, a Pade approximation of type (k, l) is also a Pade approximation of type (m, n). If ckl is the corner nonzero entry of the inﬁnite zero window’s frame, then the same is valid for all m ≥ k and n ≥ l provided that k + l ≤ m + n.

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Proof. Let k ≤ m and l ≤ n. The claim reduces to the following check: if for v l we have Akl vl = −akl , then, so long as k + l ≤ m + n < k + l + r, we also obtain

0 Amn l = −amn . v In the case of inﬁnite window, the latter holds true for all r > 0.

References [1] G.A. Baker, Jr. and P. Graves-Morris, Pade Approximants, Addison-Wesley Publishing Co., 1981. [2] A. Bultheel, M. Van Barel, Linear Algebra, Rational Approximation and Orthogonal Polynomials, Studies in Computational Mathematics, vol. 6, North-Holland, Elsevier Science, Amsterdam, 1997. [3] A. Bultheel, M. Van Barel, Pade techniques for model reduction in linear system theory: a survey, J. Comput. Appl. Math., 14, pp. 401–438 (1986). [4] I. Gohberg, A.A. Semencul, On inversion of ﬁnite-section Toeplitz matrices and their continuous analogues, Matem. Issled. (Russian), Vol.7, No. 2, pp. 201–224 (1972). [5] G. Heinig, K. Rost, Algebraic Methods for Toeplitz-like Matrices and Operators, Academie-Verlag, Berlin, 1984. [6] I.C. Iohvidov, Hankel and Toeplitz Matrices and Forms, Nauka, 1984. [7] E.M. Nikishin and V.N. Sorokin, Rational approximations and orthogonality, Nauka, 1988. [8] E.E. Tyrtyshnikov, Toeplitz Matrices, Some Analogs and Applications, OVM RAN, 1989. [9] E.E. Tyrtyshnikov, Euclidean Algorithm and Hankel Matrices, Numerical Analysis and Applied Mathematics, AIP Conference Proceedings, vol. 936, Melville, New York, pp. 27–30 (2007). [10] V.V. Voevodin and E.E. Tyrtyshnikov, Computational Processes with Toeplitz Matrices, Nauka, 1987. Eugene Tyrtyshnikov Institute of Numerical Mathematics Russian Academy of Sciences Gubkin Street, 8 Moscow 119333, Russia e-mail: [email protected]